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Intro to 3-Dimensional Solids Return to Table of Contents Slide 6 - PDF document

Mar 23, 2024 •242 likes •2.03k views

Slide 1 / 311 Slide 2 / 311 Geometry 3D Geometry 2015-10-28 www.njctl.org Slide 3 / 311 Table of Contents Intro to 3-D Solids Click on the topic to go to that section Views & Drawings of 3-D Solids Surface Area of a Prism Surface Area

  1. Slide 34 / 311 Intro to 3-D Solids A sphere is a 3-dimensional circle in that every point on the sphere is the same distance from the center. . C Similar to a circle, a sphere is named by its center point. Sphere C is the solid shown above. Slide 35 / 311 12 Which solids have 2 bases? Prism A Pyramid B Cylinder C D Cone Sphere E Slide 35 (Answer) / 311 12 Which solids have 2 bases? Prism A B Pyramid Answer Cylinder C A & C Cone D Sphere E [This object is a pull tab]

  2. Slide 36 / 311 13 Which solid has one vertex? A Prism Pyramid B Cylinder C Cone D Sphere E Slide 36 (Answer) / 311 13 Which solid has one vertex? Prism A Pyramid B Answer Cylinder C D D Cone Sphere E [This object is a pull tab] Slide 37 / 311 14 Which solid has more base edges than lateral edges? Prism A Pyramid B C Cylinder Cone D Sphere E

  3. Slide 37 (Answer) / 311 14 Which solid has more base edges than lateral edges? Prism A B Pyramid Cylinder C Answer A Cone D Sphere E [This object is a pull tab] Slide 38 / 311 15 Which solid(s) have no vertices? Prism A Pyramid B Cylinder C Cone D E Sphere Slide 38 (Answer) / 311 15 Which solid(s) have no vertices? Prism A Pyramid B C Cylinder Answer C & E Cone D Sphere E [This object is a pull tab]

  4. Slide 39 / 311 16 Which solid is formed when rotating an isosceles triangle about its altitude? A a prism B a cylinder C a pyramid D a cone E a sphere Slide 39 (Answer) / 311 16 Which solid is formed when rotating an isosceles triangle about its altitude? A a prism B a cylinder Answer C a pyramid D D a cone E a sphere [This object is a pull tab] Slide 40 / 311 Intro to 3-D Solids Euler's Theorem states that the number of faces (F), vertices (V), and edges (E) satisfy the formula F + V = E + 2 A A B C N P X Y M Q S R Z F = 5 F = 7 V = 6 V = 7 E = 9 E = 12 5 + 6 = 9 + 2 7 + 7 = 12 + 2 11 = 11 14 = 14

  5. Slide 41 / 311 Intro to 3-D Solids Example: A solid has 12 faces, 2 decagons and 10 trapezoids. How many vertices does the solid have? On their own, the 2 decagons & 10 trapezoids have 2(10) + 10(4) = 60 edges. In a 3-D solid, each side is shared by 2 polygons. Therefore, the number of edges in the solid is 60/2 = 30. V + F = E + 2 click V + 12 = 30 + 2 click V + 12 = 32 click V = 20 click Slide 41 (Answer) / 311 Intro to 3-D Solids Example: A solid has 12 faces, 2 decagons and 10 trapezoids. How many vertices does the solid have? Questioning to help address MP standards: What information do you have? (MP1) On their own, the 2 decagons & 10 trapezoids have What is this problem asking? (MP1) 2(10) + 10(4) = 60 edges. In a 3-D solid, each side is What strategies are you going to use? (MP1) How can you represent the problem with shared by 2 polygons. Therefore, the number of edges numbers and symbols? (MP2) Math Practice in the solid is 60/2 = 30. Create an equation to represent the problem. (MP2) V + F = E + 2 Would it help to draw a picture? (MP4 & MP5) click Does your answer seem reasonable? Why or V + 12 = 30 + 2 why not? (MP3) click V + 12 = 32 Can you find a shortcut to solve the problem? click How would your shortcut make the problem V = 20 easier? (MP8) click - Patterns of a prism: V = 2(# of sides of the base), E = 3(# of sides of the base) & F = (# of [This object is a pull tab] sides of the base) + 2 Slide 42 / 311 Intro to 3-D Solids Example: A solid has 9 faces, 1 octagon and 8 triangles. How many vertices does the solid have? What information do you have? 9 faces & the 2 types of faces click How are the number of edges in the 2-D faces, related to the number of edges in the polyhedron? Write a number What is the problem asking? sentence to describe this Create an equation to represent situation. the problem. V + F = E + 2 (1(8) + 8(3))/2 click click V + 9 = 16 + 2 (8 + 24)/2 click click V + 9 = 18 32/2 click click V = 9 16 edges click click

  6. Slide 42 (Answer) / 311 Intro to 3-D Solids Questions on this slide address MP standards Example: 1st Question: MP1 A solid has 9 faces, 1 octagon and 8 triangles. How 2nd set of questions: MP7 & MP4 many vertices does the solid have? 3rd set of questions: MP1 & MP2 Additional Questions to help address MP Math Practice What information do you have? standards: 9 faces & the 2 types of faces Would it help to draw a picture? (MP4 & MP5) Does your answer seem reasonable? Why or click How are the number of edges why not? (MP3) in the 2-D faces, related to the Can you find a shortcut to solve the problem? How would your shortcut make the problem number of edges in the easier? (MP8) polyhedron? Write a number What is the problem asking? - Patterns of a pyramid: V = (# of sides of the sentence to describe this Create an equation to represent base) + 1, E = 2(# of sides of the base) & situation. the problem. F = (# of sides of the base) + 1 [This object is a pull tab] V + F = E + 2 (1(8) + 8(3))/2 click click V + 9 = 16 + 2 (8 + 24)/2 click click V + 9 = 18 32/2 click click V = 9 16 edges click click Slide 43 / 311 17 A solid has 10 faces, one of them being a nonagon and 9 triangles. How many vertices does it have? 8 A 9 B 10 C D 18 Slide 43 (Answer) / 311 17 A solid has 10 faces, one of them being a nonagon and 9 triangles. How many vertices does it have? 8 A B 9 10 C Answer C 18 D [This object is a pull tab]

  7. Slide 44 / 311 18 A solid has 12 faces, all of them being pentagons. How many vertices does it have? A 30 20 B 15 C 10 D Slide 44 (Answer) / 311 18 A solid has 12 faces, all of them being pentagons. How many vertices does it have? 30 A 20 B Answer B 15 C D 10 [This object is a pull tab] Slide 45 / 311 19 A solid has 8 faces, all of them being triangles. How many vertices does it have? 24 A B 12 8 C 6 D

  8. Slide 45 (Answer) / 311 19 A solid has 8 faces, all of them being triangles. How many vertices does it have? A 24 12 B Answer 8 C D 6 D [This object is a pull tab] Slide 46 / 311 Intro to 3-D Solids Cross-Section A cross-section is the locus of points of the intersection of a plane and a 3-D solid. Slide 47 / 311 Cross-Section Think about it as if the plane were a knife and you were cutting the shape, what would the cut look like? Circle Ellipse Parabola (with the inner section shaded)

  9. Slide 48 / 311 Cross-Section Cross-sections of a surface are a 2-dimensional figure. Cross-sections of a solid are a 2-dimensional figure and its interior. The top can be removed to see the cross section. (Try it out) Slide 49 / 311 20 What is the locus of points (cross-section) of a cube and a plane perpendicular to the base and parallel to the non-intersecting sides? square A rectangle B trapezoid C hexagon D E rhombus parallelogram F triangle G circle H Slide 49 (Answer) / 311 20 What is the locus of points (cross-section) of a cube and a plane perpendicular to the base and parallel to the non-intersecting sides? square A rectangle B A Answer C trapezoid hexagon D rhombus E parallelogram F [This object is a pull tab] triangle G H circle

  10. Slide 50 / 311 21 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane perpendicular to the base and parallel to the non- intersecting sides? A 72 sq inches B 144 sq inches C 187.06 sq inches 12 in. D 203.65 sq inches Slide 50 (Answer) / 311 21 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane perpendicular to the base and parallel to the non- intersecting sides? A 72 sq inches B B 144 sq inches Answer 12 in. C 187.06 sq inches 12 in. 12 in. D 203.65 sq inches [This object is a pull tab] Slide 51 / 311 22 What is the locus of points of a cube and a plane that contains the diagonal of the base and is perpendicular to the base? square A rectangle B C trapezoid hexagon D rhombus E parallelogram F triangle G H circle

  11. Slide 51 (Answer) / 311 22 What is the locus of points of a cube and a plane that contains the diagonal of the base and is perpendicular to the base? square A B rectangle B trapezoid C Answer hexagon D rhombus E parallelogram F [This object is a pull tab] G triangle circle H Slide 52 / 311 23 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane that contains the diagonal of the base and is perpendicular to the base? A 72 sq inches B 144 sq inches C 187.06 sq inches 12 in. D 203.65 sq inches Slide 52 (Answer) / 311 23 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane that contains the diagonal of the base and is perpendicular to the base? D A 72 sq inches 12√2 in. Answer B 144 sq inches 12 in. C 187.06 sq inches 12 in. D 203.65 sq inches [This object is a pull tab]

  12. Slide 53 / 311 24 What is the locus of points of a cube and a plane that contains the diagonal of the base but does not intersect the opposite base? square A B rectangle trapezoid C hexagon D rhombus E parallelogram F G triangle circle H Slide 53 (Answer) / 311 24 What is the locus of points of a cube and a plane that contains the diagonal of the base but does not intersect the opposite base? square A G rectangle B Answer trapezoid C hexagon D E rhombus parallelogram F [This object is a pull tab] triangle G circle H Slide 54 / 311 25 What is the locus of points of a cube and a plane that intersects all of the faces? square A rectangle B trapezoid C hexagon D rhombus E parallelogram F triangle G circle H

  13. Slide 54 (Answer) / 311 25 What is the locus of points of a cube and a plane that intersects all of the faces? square A rectangle B D trapezoid C Answer hexagon D E rhombus parallelogram F triangle G [This object is a pull tab] circle H Slide 55 / 311 Views & Drawings of 3-D Solids Return to Table of Contents Slide 56 / 311 Views & Drawings Isometric drawings are drawings that look 3-D & are created on a grid of dots using 3 axes that intersect to form 120° & 60° angles.

  14. Slide 57 / 311 Views & Drawings Example: Create an Isometric drawing of a cube. Slide 57 (Answer) / 311 Views & Drawings Example: Create an Isometric drawing of a cube. OR Answer [This object is a pull tab] Slide 58 / 311 Views & Drawings An Orthographic projection is a 2-D drawing that shows the different viewpoints of an object, usually from the front, top & side. Each drawing depends on your position relative to the figure. Top (from front) Front Side

  15. Slide 59 / 311 Views & Drawings Consider these three people viewing a pyramid: Slide 60 / 311 Consider these three people viewing a pyramid: The orange person is standing in front of a face, so their view is a triangle. Slide 61 / 311 Consider these three people viewing a pyramid: The green person is standing in front of a lateral edge, so from their view they can see 2 faces.

  16. Slide 62 / 311 Consider these three people viewing a pyramid: The purple person is flying over and can see the four lateral faces. Slide 63 / 311 26 Given the surface shown, what would be the view from point A? a Rectangle A B a Square a Circle C a Pentagon D A (front) a Triangle E a Parallelogram F G a Hexagon a Trapezoid H Slide 63 (Answer) / 311 26 Given the surface shown, what would be the view from point A? a Rectangle A a Square B a Circle C Answer E a Pentagon D A (front) E a Triangle a Parallelogram F [This object is a pull tab] a Hexagon G a Trapezoid H

  17. Slide 64 / 311 27 Given the surface shown, what would be the view from point A? a Rectangle A A (top) a Square B a Circle C D a Pentagon a Triangle E a Parallelogram F a Hexagon G a Trapezoid H Slide 64 (Answer) / 311 27 Given the surface shown, what would be the view from point A? a Rectangle A A (top) B a Square a Circle Answer C A a Pentagon D a Triangle E a Parallelogram F [This object is a pull tab] G a Hexagon a Trapezoid H Slide 65 / 311 28 Given the surface shown, what would be the view from point A? A (top) a Rectangle A a Square B a Circle C a Pentagon D E a Triangle a Parallelogram F a Hexagon G right square prism a Trapezoid H

  18. Slide 65 (Answer) / 311 28 Given the surface shown, what would be the view from point A? A (top) a Rectangle A a Square B Answer B a Circle C D a Pentagon a Triangle E a Parallelogram F [This object is a pull tab] a Hexagon G right square prism a Trapezoid H Slide 66 / 311 29 Given the surface shown, what would be the view from point A? right square prism a Rectangle A B a Square a Circle C a Pentagon D a Triangle E a Parallelogram F A (front) G a Hexagon a trapezoid H Slide 66 (Answer) / 311 29 Given the surface shown, what would be the view from point A? right square prism a Rectangle A a Square B a Circle C Answer A a Pentagon D E a Triangle a Parallelogram F A [This object is a pull tab] (front) a Hexagon G a trapezoid H

  19. Slide 67 / 311 30 Given the surface shown, what would be the view from point A? a Rectangle A a Square B a Circle C D a Pentagon A (front) a Triangle E a Parallelogram F a Hexagon G a Trapezoid H Slide 67 (Answer) / 311 30 Given the surface shown, what would be the view from point A? a Rectangle A B a Square a Circle C Answer E a Pentagon D A (front) a Triangle E a Parallelogram F [This object is a pull tab] G a Hexagon a Trapezoid H Slide 68 / 311 31 Given the surface shown, what would be the view from point A? A (above) a Rectangle A a Square B a Circle C a Pentagon D E a Triangle a Parallelogram F a Hexagon G a Trapezoid H

  20. Slide 68 (Answer) / 311 31 Given the surface shown, what would be the view from point A? A (above) a Rectangle A a Square B a Circle C Answer C D a Pentagon a Triangle E a Parallelogram F [This object is a pull tab] a Hexagon G a Trapezoid H Slide 69 / 311 32 Given the surface shown, what would be the view from point A? A (above) a Rectangle A B a Square a Circle C a Pentagon D a Triangle E a Parallelogram F G a Hexagon a Trapezoid H Slide 69 (Answer) / 311 32 Given the surface shown, what would be the view from point A? A (above) a Rectangle A a Square B Answer a Circle C C a Pentagon D E a Triangle a Parallelogram F [This object is a pull tab] a Hexagon G a Trapezoid H

  21. Slide 70 / 311 33 Given the surface shown, what would be the view from point A? a Rectangle A a Square B a Circle C A D a Pentagon (front) a Triangle E a Parallelogram F a Hexagon G a Trapezoid H Slide 70 (Answer) / 311 33 Given the surface shown, what would be the view from point A? a Rectangle A B a Square Answer a Circle C A A a Pentagon D (front) a Triangle E a Parallelogram F [This object is a pull tab] G a Hexagon a Trapezoid H Slide 71 / 311 34 Given the surface shown, what would be the view from point A? sphere a Rectangle A a Square B a Circle C a Pentagon D A E a Triangle a Parallelogram F a Hexagon G a Trapezoid H

  22. Slide 71 (Answer) / 311 34 Given the surface shown, what would be the view from point A? sphere a Rectangle A a Square B Answer C a Circle C D a Pentagon A a Triangle E a Parallelogram F [This object is a pull tab] a Hexagon G a Trapezoid H Slide 72 / 311 Views & Drawings C (Looking down from above) What would the view be like from each position? A B Slide 73 / 311 Views & Drawings What would the view be like from each position? From A, how many columns of blocks are visible? - 3 columns Click to A reveal How tall is each column? - first one is 4 high - second & third columns are each 2 blocks high Click to reveal

  23. Slide 73 (Answer) / 311 Views & Drawings Questions on this slide address MP standards 1st Question: MP1 2nd Question: MP1 Math Practice What would the view be like from each position? Additional Questions to help address MP standards: Does your answer seem reasonable? Why or why not? (MP3) Why does your final drawing make sense? (MP4) From A, how many columns of blocks are visible? - 3 columns [This object is a pull tab] A Click to reveal How tall is each column? - first one is 4 high - second & third columns are each 2 blocks high Click to reveal Slide 74 / 311 Views & Drawings What would the view be like from each position? From B, how many columns of blocks are visible? B - 2 columns Click to reveal How tall is each column? - left one is 3 high - right one is 4 high Click to reveal Slide 74 (Answer) / 311 Views & Drawings Questions on this slide address MP standards 1st Question: MP1 2nd Question: MP1 Math Practice What would the view be Additional Questions to help address MP like from each position? standards: Does your answer seem reasonable? Why or why not? (MP3) Why does your final drawing make sense? (MP4) From B, how many columns of blocks are visible? B - 2 columns [This object is a pull tab] Click to reveal How tall is each column? - left one is 3 high - right one is 4 high Click to reveal

  24. Slide 75 / 311 (Looking down C Views & Drawings from above) What would the view be like from each position? From C, how many columns of blocks are visible? - 3 columns Click to reveal How tall is each column? - all of them are 2 blocks high Click to reveal Slide 75 (Answer) / 311 C (Looking down Views & Drawings from above) Questions on this slide address MP standards 1st Question: MP1 What would the view be 2nd Question: MP1 Math Practice like from each position? Additional Questions to help address MP standards: Does your answer seem reasonable? Why or why not? (MP3) From C, how many columns of Why does your final drawing make sense? blocks are visible? (MP4) - 3 columns Click to reveal How tall is each column? [This object is a pull tab] - all of them are 2 blocks high Click to reveal Slide 76 / 311 Views & Drawings Above Draw the 3 views. Front Side Move for Answer Front View Side View Top View

  25. Slide 76 (Answer) / 311 Views & Drawings Above Draw the 3 views. Questions to help address MP standards: From each viewpoint, how many columns of blocks are visible? (MP1) Math Practice For this viewpoint, how tall is each column? (MP1) Front Does your answer seem reasonable? Why or Side why not? (MP3) Why does your final drawing make sense? (MP4) Move for Answer Front View [This object is a pull tab] Side View Top View Slide 77 / 311 Views & Drawings Draw the 3 views. Above Front Side Move for Answer Side Above Front Slide 77 (Answer) / 311 Views & Drawings Draw the 3 views. Above Questions to help address MP standards: From each viewpoint, how many columns of blocks are visible? (MP1) Math Practice For this viewpoint, how tall is each column? (MP1) Front Does your answer seem reasonable? Why or Side why not? (MP3) Why does your final drawing make sense? (MP4) Move for Answer Side Above Front [This object is a pull tab]

  26. Slide 78 / 311 Views & Drawings Here are 3 views of a solid, draw a 3-dimensional representation. L R F Side Top Front Move for Answer Slide 78 (Answer) / 311 Views & Drawings Here are 3 views of a solid, draw a 3-dimensional representation. Questions to help address MP standards: What information are you given? (MP1) L R What strategies are you going to use? (MP1) Math Practice Can you guess & check? (MP5) F - Referring to drawing a "guess diagram" for the Side Top Front solid & "checking" to see if it works Does your answer seem reasonable? Why or why not? (MP3) Move for Answer Why does your final drawing make sense? (MP4) [This object is a pull tab] Slide 79 / 311 Views & Drawings Here are 3 views of a solid, draw a 3-dimensional representation. L R Front Side F Top Move for Answer

  27. Slide 79 (Answer) / 311 Views & Drawings Here are 3 views of a solid, draw a 3-dimensional representation. R L Questions to help address MP standards: What information are you given? (MP1) Front What strategies are you going to use? (MP1) Side Math Practice Can you guess & check? (MP5) F - Referring to drawing a "guess diagram" for the Top solid & "checking" to see if it works Does your answer seem reasonable? Why or Move for Answer why not? (MP3) Why does your final drawing make sense? (MP4) [This object is a pull tab] Slide 80 / 311 Surface Area of a Prism Return to Table of Contents Slide 81 / 311 Net A Net is a 2-dimensional shape that folds into a 3-dimensional figure. The Net shows all of the faces of the surface. Shown is the net of a right rectangular prism. 4 6 12 12 4 6 4 6 4 6

  28. Slide 81 (Answer) / 311 Net A Net is a 2-dimensional shape that folds into a 3-dimensional figure. The PARCC Reference sheet for the HS level does NOT contain any The Net shows all of the faces of the surface. formulas to calculate the Surface Teacher's Note Area (reference sheet is linked to Shown is the net of a right rectangular prism. this pull tab - just click on it to download it). Encourage students to either memorize future formulas, 4 or draw the net each time so that 6 they can break down the solid into smaller 2-D shapes. [This object is a pull tab] 12 12 6 4 6 4 4 6 Slide 82 / 311 Net The net shown is a right triangular prism. The lateral faces are rectangles. The bases are on opposite sides of the rectangles, although they do not need to be on the same rectangle. Slide 83 / 311 Net The nets shown are for the same right triangular prism.

  29. Slide 84 / 311 Nets Nets of oblique prisms have parallelograms as lateral faces. Slide 85 / 311 Rectangular Prisms cube H H H w w w ℓ ℓ ℓ Slide 85 (Answer) / 311 Rectangular Prisms To avoid confusion with the cube Teacher's Note "heights" when calculating the H surface area of a triangular prism, the height of the prism has been H assigned "H". The triangular height H will be "h" starting on slide #98 w w w ℓ ℓ ℓ [This object is a pull tab]

  30. Slide 86 / 311 Rectangular Prisms Base Base height height Base Base A prism has 2 bases. The base of a rectangular prism is a rectangle. The height of the prism is the length between the two bases. Slide 87 / 311 Rectangular Prisms The Surface Area of a figure is the total amount of area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift). The Surface Area of a figure is the sum of the areas of each side of the figure. Area Top Area Area Back Side Side Area Area Area Area Area Area Front Area Bottom Area Area Slide 88 / 311 Finding the Surface Area of a Rectangular Prism H w ℓ Area of the Top = ℓ x w Area of the Front = ℓ x H Area of the Bottom = ℓ x w Area of the Back = ℓ x H Area of Left Side = w x H Area of Right Side = w x H The Surface Area is the sum of all the areas S.A. = ℓw + ℓw + ℓH + ℓH + wH + wH S.A. = 2 ℓw + 2 ℓH + 2wH

  31. Slide 88 (Answer) / 311 Finding the Surface Area of a Rectangular Prism H w To avoid confusion with the ℓ Teacher's Note "heights" when calculating the Area of the Top = ℓ x w Area of the Front = ℓ x H surface area of a triangular prism, the height of the prism has been Area of the Bottom = ℓ x w assigned "H". The triangular height Area of the Back = ℓ x H will be "h" starting on slide #98 Area of Left Side = w x H Area of Right Side = w x H [This object is a pull tab] The Surface Area is the sum of all the areas S.A. = ℓw + ℓw + ℓH + ℓH + wH + wH S.A. = 2 ℓw + 2 ℓH + 2wH Slide 89 / 311 Finding the Surface Area of a Rectangular Prism Example: Find the surface area of the prism 3 4 7 Area of Top & Bottom Area of Right & Left A = 7(4) = 28u 2 A = 3(4) = 12 u 2 Click Click Area of Front & Back A = 3(7) = 21 u 2 Click Total Surface Area = 2(28) + 2(12) + 2(21) Click = 56 + 24 + 42 = 122 units 2 Click Slide 89 (Answer) / 311 Finding the Surface Area of a Rectangular Prism Example: Find the surface area of the prism Questions to help address MP standards: 3 What information are you given? (MP1) What is the problem asking? (MP1) Math Practice How can you represent the problem with 4 symbols and numbers? (MP2) What tools do you need? (MP5) 7 Area of Top & Bottom Area of Right & Left Can you do this mentally? (MP5) What labels could you use? (MP6) A = 7(4) = 28u 2 A = 3(4) = 12 u 2 Click Click Area of Front & Back A = 3(7) = 21 u 2 [This object is a pull tab] Click Total Surface Area = 2(28) + 2(12) + 2(21) Click = 56 + 24 + 42 = 122 units 2 Click

  32. Slide 90 / 311 35 What is the total surface area, in square units? 5 9 4 Slide 90 (Answer) / 311 35 What is the total surface area, in square units? 5 SA = 2(4)(5) + 2(4)(9) + 2(5)(9) 9 Answer SA = 40 + 72 + 90 SA = 202 sq. units 4 [This object is a pull tab] Slide 91 / 311 36 What is the total surface area, in square units? 8 8 8

  33. Slide 91 (Answer) / 311 36 What is the total surface area, in square units? SA = 6(8)(8) 8 Answer SA = 6(64) SA = 384 sq. units 8 8 [This object is a pull tab] Slide 92 / 311 37 Troy wants to build a cube out of straws. The cube is to have a total surface area of 96 in 2 , what is the total length of the straws, in inches? Slide 92 (Answer) / 311 37 Troy wants to build a cube out of straws. The cube is to have a total surface area of 96 in 2 , what is the total length of the straws, in inches? 96 = 6(x)(x) Answer 96 = 6x 2 16 = x 2 x = 4 in. [This object is a pull tab]

  34. Slide 93 / 311 Another Way of Looking at Surface Area S.A. = 2B + PH The Surface Area is the sum of the areas of the 2 Bases plus the Lateral Area (Perimeter of the base, P, times the height of the prism, H) The Lateral Area is the area of the Lateral Surface. The Lateral Surface is the part that wraps around the middle of the figure (in between the two bases). e s a B Base Lateral Surface Base e s a B Slide 93 (Answer) / 311 Another Way of Looking at Surface Area S.A. = 2B + PH The Surface Area is the sum of the areas of the 2 Bases plus the Lateral Area (Perimeter of the base, P, times the height of the prism, H) The Lateral Area is the area of the Lateral Surface. The Lateral To avoid confusion with the Surface is the part that wraps around the middle of the figure (in Teacher's Note "heights" when calculating the between the two bases). surface area of a triangular prism, e s a the height of the prism has been B assigned "H". The triangular height Base will be "h" starting on slide #98 Lateral Surface [This object is a pull tab] Base e s a B Slide 94 / 311 Rectangular Prisms Another formula for Surface Area of a right prism: S.A. = 2B + PH B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism Base H Base w ℓ S.A. = 2B + PH S.A. = 2 ℓw + (2 ℓ +2w)H S.A. = 2 ℓw + 2 ℓH + 2wH

  35. Slide 95 / 311 Rectangular Prisms Another formula for Surface Area of a right prism: S.A. = 2B + PH B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism Base H Base w ℓ In the surface area formula, 2B is the sum of the area of the 2 bases. The area of lateral faces or What does PH represent? Lateral Area Click Slide 96 / 311 38 If the base of the prism is 12 by 6, what is the lateral area, in sq ft? 4 ft 6 ft 12 ft Slide 96 (Answer) / 311 38 If the base of the prism is 12 by 6, what is the lateral area, in sq ft? LA = PH 4 ft Answer P = 2(12) + 2(6) = 36 ft LA = 36(4) 6 ft LA = 144 sq. ft 12 ft [This object is a pull tab]

  36. Slide 97 / 311 39 The surface area of the rectangular prism is : 24 sq ft A 4 ft 144 sq ft B 6 ft 288 sq ft C 12 ft 48 sq ft D 72 sq ft E Slide 97 (Answer) / 311 39 The surface area of the rectangular prism is : A 24 sq ft 4 ft C B 144 sq ft 6 ft Answer 288 sq ft SA = 2B + PH C 12 ft SA = 2(12)(6) + 36(4) 48 sq ft D SA = 144 + 144 sq ft 72 sq ft E [This object is a pull tab] Slide 98 / 311 40 If 7 by 6 is base of the prism, what is the lateral area, in sq units? 7 6 9

  37. Slide 98 (Answer) / 311 40 If 7 by 6 is base of the prism, what is the lateral area, in sq units? LA = PH Answer P = 2(6) + 2(7) = 26 units 7 LA = 26(9) 6 LA = 234 sq. units 9 [This object is a pull tab] Slide 99 / 311 41 What is the total square units of the surface area? 7 6 9 Slide 99 (Answer) / 311 41 What is the total square units of the surface area? SA = 2B + PH 7 Answer SA = 2(6)(7) + 26(9) 6 SA = 84 + 234 9 SA = 318 units 2 [This object is a pull tab]

  38. Slide 100 / 311 42 Find the value of y, if the lateral area is 144 sq units, and y by 6 is the base. y 6 8 Slide 100 (Answer) / 311 42 Find the value of y, if the lateral area is 144 sq units, and y by 6 is the base. LA = PH y P = 2(6) + 2y = 12 + 2y Answer 144 = (12 + 2y)8 6 8 144 = 96 + 16y 48 = 16y 3 = y [This object is a pull tab] Slide 101 / 311 43 What is the value of the missing variable if the surface area is 350 sq. ft. 7 ft A 8.3 ft B 10 ft 12 ft C 15 ft D 5 ft x ft

  39. Slide 101 (Answer) / 311 43 What is the value of the missing variable if the surface area is 350 sq. ft. B 7 ft A SA = 2B + PH 8.3 ft B P = 2(5) + 2x = 10 + 2x Answer 10 ft 12 ft B = 5x C 350 = 2(5x) + (10 + 2x)10 15 ft D 350 = 10x + 100 + 20x 350 = 30x + 100 250 = 30x 5 ft x = 8.3 ft [This object is a pull tab] x ft Slide 102 / 311 44 Sharon was invited to Maria's birthday party. For a present, she purchased an iHome (a clock radio for an iPod or iPhone) which is contained in a box that measures 7 inches in length, 5 inches in width, and 4 inches in height. How much wrapping paper does Sharon need to wrap Maria's present? Slide 102 (Answer) / 311 44 Sharon was invited to Maria's birthday party. For a present, she purchased an iHome (a clock radio for an iPod or iPhone) which is contained in a box that measures 7 inches in length, 5 inches in width, and 4 inches in height. How much wrapping paper does Sharon need to wrap Maria's present? SA = 2(5)(7) + 2(5)(4) + 2(7)(4) Answer = 70 + 40 + 56 SA = 166 sq. inches [This object is a pull tab]

  40. Slide 103 / 311 Other Prisms Slide 104 / 311 Other Prisms base base base height height base base base base height base height A Prism has 2 Bases The Base of a Prism matches the first word in the name of the prism. e.g. the Base of a Triangular Prism is a Triangle The Height of the Prism is the length between the two bases Slide 105 / 311 Other Prisms The Surface Area of a figure is the total amount of Area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift). The Surface Area of a figure is the sum of the areas of each side of the figure Area Area Area Area Area Area Area Area Area Area Triangular Prism Net of the Triangular Prism

  41. Slide 106 / 311 Finding the Surface Area of a Right Prism Surface Area: S.A. = 2B + PH B = Area of the triangular base = ½bh P = Perimeter of the triangular base = a + b + c H = Height of the prism Lateral Area = PH = (a + b + c)H The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases. a c base c a b h a c Prism's Lateral Surface base H H b height b B = ½ bh P = a + b + c Note: The formula above will work for any right prism. Slide 107 / 311 Other Prisms Example: Find the lateral area and surface area of the right triangular prism. 10 6 11 Since it has a base that is a right triangle, we need to find the base of the triangle using Pythagorean Theorem. 6 2 + b 2 = 10 2 36 + b 2 = 100 b 2 = 64 b = 8 units Next, calculate the perimeter of your base. P = 6 + 8 + 10 = 24 units Use this to find the Lateral Area LA = PH = 24(11) = 264 units 2 Slide 107 (Answer) / 311 Other Prisms Questions to help address MP standards: What information are you given? (MP1) Example: Find the lateral area and surface area of the What is the problem asking? (MP1) right triangular prism. How can you represent the problem with Math Practice symbols and numbers? (MP2) 10 What tools do you need? (MP5) Can you do this mentally? (MP5) - Referring to the Pythagorean Triple Can you find a shortcut to solve the problem? 6 How would the shortcut make the problem 11 easier? (MP8) - Referring to the Pythagorean Triple What labels could you use? (MP6) Since it has a base that is a right triangle, we need to find the base of the triangle using Pythagorean Theorem. [This object is a pull tab] 6 2 + b 2 = 10 2 36 + b 2 = 100 b 2 = 64 b = 8 units Next, calculate the perimeter of your base. P = 6 + 8 + 10 = 24 units Use this to find the Lateral Area LA = PH = 24(11) = 264 units 2

  42. Slide 108 / 311 Other Prisms Example: Find the lateral area and surface area of the right triangular prism. 10 6 11 Then, calculate the area of your base, B B = (1/2)(8)(6) = 24 units 2 Finally, calculate your Surface Area. SA = 2B + PH SA = 2(24) + (24)(11) SA = 48 + 264 = 312 units 2 Slide 108 (Answer) / 311 Other Prisms Questions to help address MP standards: Example: Find the lateral area and surface area of the right What information are you given? (MP1) triangular prism. What is the problem asking? (MP1) Math Practice How can you represent the problem with 10 symbols and numbers? (MP2) What tools do you need? (MP5) Can you do this mentally? (MP5) What labels could you use? (MP6) 6 11 Then, calculate the area of your base, B [This object is a pull tab] B = (1/2)(8)(6) = 24 units 2 Finally, calculate your Surface Area. SA = 2B + PH SA = 2(24) + (24)(11) SA = 48 + 264 = 312 units 2 Slide 109 / 311 Other Prisms Example: Find the lateral area and surface area of the triangular prism. 9 9 12 9 Since it has a base that is an equilateral triangle, we need to find the height of the triangle using Pythagorean Theorem or the 30-60-90 Triangle Theorem. 4.5 2 + b 2 = 9 2 20.25 + b 2 = 81 b 2 = 60.75 b = 4.5√3 units = 7.79 units Next, calculate the perimeter of your base. P = 9 + 9 + 9 = 27 units Use this to find the Lateral Area LA = PH = 27(12) = 324 units 2

  43. Slide 109 (Answer) / 311 Other Prisms Questions to help address MP standards: What information are you given? (MP1) Example: Find the lateral area and surface area of the What is the problem asking? (MP1) triangular prism. How can you represent the problem with Math Practice 9 9 symbols and numbers? (MP2) What tools do you need? (MP5) Can you do this mentally? (MP5) - Referring to the 30-60-90 triangle 12 Can you find a shortcut to solve the problem? How would the shortcut make the problem easier? (MP8) 9 - Referring to the 30-60-90 triangle Since it has a base that is an equilateral triangle, we need to find the What labels could you use? (MP6) height of the triangle using Pythagorean Theorem or the 30-60-90 [This object is a pull tab] Triangle Theorem. 4.5 2 + b 2 = 9 2 20.25 + b 2 = 81 b 2 = 60.75 b = 4.5√3 units = 7.79 units Next, calculate the perimeter of your base. P = 9 + 9 + 9 = 27 units Use this to find the Lateral Area LA = PH = 27(12) = 324 units 2 Slide 110 / 311 Other Prisms Example: Find the lateral area and surface area of the triangular prism. 9 9 12 9 Then, calculate the area of your base, B B = (1/2)(9)(4.5√3) = 20.25√3 units 2 = 35.07 units 2 Finally, calculate your Surface Area. SA = 2B + PH SA = 2(35.07) + (27)(12) SA = 70.14 + 324 = 394.14 units 2 Slide 110 (Answer) / 311 Other Prisms Example: Find the lateral area and surface area of the Questions to help address MP standards: triangular prism. What information are you given? (MP1) 9 9 What is the problem asking? (MP1) Math Practice How can you represent the problem with symbols and numbers? (MP2) What tools do you need? (MP5) 12 Can you do this mentally? (MP5) What labels could you use? (MP6) 9 Then, calculate the area of your base, B B = (1/2)(9)(4.5√3) = 20.25√3 units 2 = 35.07 units 2 [This object is a pull tab] Finally, calculate your Surface Area. SA = 2B + PH SA = 2(35.07) + (27)(12) SA = 70.14 + 324 = 394.14 units 2

  44. Slide 111 / 311 45 The height of the triangular prism below is 11 ft, the height of the base is 3 ft, and the triangular base is an isosceles triangle. Find the surface area. A 88 sq ft B 132 sq ft 5 ft C 198 sq ft 3 ft 11 ft D 222 sq ft Slide 111 (Answer) / 311 45 The height of the triangular prism below is 11 ft, the height of the base is 3 ft, and the triangular base is an D isosceles triangle. Find the surface area. (1/2) of the base in the triangle is 4...3-4-5 Pyth. Triple, so the A 88 sq ft base of the triangle is 8. Answer B 132 sq ft P = 5 + 5 + 8 = 18 5 ft C 198 sq ft B = (1/2)(8)(3) = 12 3 ft 11 ft D 222 sq ft SA = 2(12) + 18(11) SA = 24 + 198 SA = 222 sq ft [This object is a pull tab] Slide 112 / 311 46 The height of the triangular prism below is 3, and the triangular base is an equilateral triangle. Find the surface area. A 64 sq ft B 127.43 sq ft C 72 sq ft 3 ft D 55.43 sq ft 8 ft

  45. Slide 112 (Answer) / 311 46 The height of the triangular prism below is 3, and the triangular base is an equilateral triangle. Find the surface area. B The height of the triangular base is 4√3 = 6.93...30-60-90 A 64 sq ft triangle. Answer B 127.43 sq ft 30 o 8 P = 8(3) = 24 C 72 sq ft h B = (1/2)(8)(4√3) = 16√3 3 ft 60 o D 4 4 55.43 sq ft SA = 2(16√3) + 24(3) 8 ft SA = 32√3 + 72 SA = 127.43 sq ft [This object is a pull tab] Slide 113 / 311 47 Find the lateral area of the right prism. 5 6 5 Slide 113 (Answer) / 311 47 Find the lateral area of the right prism. hypotenuse of the right triangle is 5√2...45-45-90 triangle Answer P = 5 + 5 + 5√2 = 10 + 5√2 = = 17.07 5 LA = (10 + 5√2)6 LA = 60 + 30√2 6 LA = 102.43 sq units 5 [This object is a pull tab]

  46. Slide 114 / 311 Finding the Surface Area of a Right Prism Surface Area : S.A. = 2B + PH B = Area of the regular hexagonal base = ½aP - a is the apothem of the regular base P = Perimeter of the base = b + c + d + e + f + g H = Height of the prism = H Lateral Area = PH = (b + c + d + e + f + g)H d e c base b f e d c b g f Prism's H height base P = b + c + d + e + f + g a B = ½ aP Slide 115 / 311 Finding the Surface Area of a Right Prism The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases. d e c base b f e d c b g f Prism's H height base P = b + c + d + e + f + g a B = ½ aP Slide 116 / 311 Other Prisms Example: Find the lateral area and surface area of the regular hexagonal prism. Because the base is a regular polygon, we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle. 360 7 in = 60° = central angle 6 Click 60 = 30° = top angle of the triangle. 2 8 in Click 30° a 4 in. Click

  47. Slide 116 (Answer) / 311 Other Prisms Example: Find the lateral area and surface area of the When calculating the surface area and/or regular hexagonal prism. volume of regular polygonal prisms/ Because the base is a regular polygon, pyramids, students could also find the we need to calculate the apothem. To measurement of the 2 congruent base Teacher's Note angles using the Triangle Sum Theorem. begin, figure out the central angle & top angle in the triangle. Questions to help address MP standards: What information are you given? (MP1) 360 7 in = 60° = central angle What is the problem asking? (MP1) 6 Click How can you represent the problem with symbols and numbers? (MP2) 60 Can you do this mentally? (MP5) = 30° = top angle of the triangle. 2 What labels could you use? (MP6) 8 in Click [This object is a pull tab] 30° a 4 in. Click Slide 117 / 311 Other Prisms Example: Find the lateral area and surface area of the regular hexagonal prism. Next find the apothem using trigonometry, or special right triangles (if it applies). 4 7 in tan 30 = a Click atan30 = 4 8 in tan30 tan30 Click a = 4√3 = 6.93 in. Click Slide 117 (Answer) / 311 Other Prisms Questions to help address MP standards: Example: Find the lateral area and surface area of the What information are you given? (MP1) What is the problem asking? (MP1) regular hexagonal prism. How can you represent the problem with symbols and numbers? (MP2) Math Practice Next find the apothem using What tools do you need? (MP5) trigonometry, or special right Can you do this mentally? (MP5) triangles (if it applies). What labels could you use? (MP6) How is right triangle trigonometry used to 4 7 in calculate the segment lengths of regular tan 30 = a polygons? (MP7) Click Can you find a shortcut to solve the problem? How would your shortcut make the problem atan30 = 4 easier? (MP8) 8 in tan30 tan30 - Referring to 30-60-90 triangle [This object is a pull tab] Click a = 4√3 = 6.93 in. Click

  48. Slide 118 / 311 Other Prisms Example: Find the lateral area and surface area of the regular hexagonal prism. Next, calculate the perimeter of your base. P = 8(6) = 48 in Click Use this to find the Lateral Area LA = PH = 48(7) = 336 in 2 7 in Click Then, calculate the area of your base, B B = (1/2)aP = (1/2)(4√3)(48) = 8 in Click Click 96√3 in 2 = 166.28in 2 Click Click Finally, calculate your Surface Area. SA = 2B + PH Click SA = 2(166.28) + (48)(7) Click SA = 332.56 + 336 = 668.56 in 2 Click Slide 118 (Answer) / 311 Other Prisms Example: Find the lateral area and surface area of the regular hexagonal prism. Next, calculate the perimeter of your base. Questions to help address MP standards: What information are you given? (MP1) P = 8(6) = 48 in What is the problem asking? (MP1) Math Practice Click How can you represent the problem with Use this to find the Lateral Area symbols and numbers? (MP2) LA = PH = 48(7) = 336 in 2 What tools do you need? (MP5) 7 in Can you do this mentally? (MP5) Click Then, calculate the area of your base, B What labels could you use? (MP6) B = (1/2)aP = (1/2)(4√3)(48) = 8 in Click Click 96√3 in 2 = 166.28in 2 Click Click [This object is a pull tab] Finally, calculate your Surface Area. SA = 2B + PH Click SA = 2(166.28) + (48)(7) Click SA = 332.56 + 336 = 668.56 in 2 Click Slide 119 / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. Because the base is a regular polygon, 6 ft we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle. 360 = 72° = central angle 10 ft 5 Click 72 = 36° = top angle of the triangle. 2 Click The base is a regular pentagon. 36° a 3 in. Click

  49. Slide 119 (Answer) / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. Because the base is a regular polygon, When calculating the surface area and/or 6 ft we need to calculate the apothem. To volume of regular polygonal prisms/ begin, figure out the central angle & top pyramids, students could also find the measurement of the 2 congruent base angle in the triangle. Teacher's Note angles using the Triangle Sum Theorem. 360 10 ft = 72° = central angle Questions to help address MP standards: 5 Click What information are you given? (MP1) What is the problem asking? (MP1) 72 How can you represent the problem with = 36° = top angle of the triangle. 2 symbols and numbers? (MP2) Click Can you do this mentally? (MP5) The base is a regular What labels could you use? (MP6) pentagon. [This object is a pull tab] 36° a 3 in. Click Slide 120 / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. 6 ft Next find the apothem using trigonometry, or special right triangles (if it applies). 10 ft 3 tan 36 = a Click atan36 = 3 tan36 tan36 Click The base is a regular a = 4.13 in. pentagon. Click Slide 120 (Answer) / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. Questions to help address MP standards: Next find the apothem using 6 ft What information are you given? (MP1) Math Practice What is the problem asking? (MP1) trigonometry, or special right How can you represent the problem with triangles (if it applies). symbols and numbers? (MP2) What tools do you need? (MP5) How is right triangle trigonometry used to 10 ft 3 tan 36 = calculate the segment lengths of regular a polygons? (MP7) Click What labels could you use? (MP6) atan36 = 3 tan36 tan36 Click [This object is a pull tab] The base is a regular a = 4.13 in. pentagon. Click

  50. Slide 121 / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. Next, calculate the perimeter of your base. 6 ft P = 5(6) = 30 in Click Use this to find the Lateral Area LA = PH = 30(10) = 300 in 2 10 ft Click Then, calculate the area of your base, B B = (1/2)aP = (1/2)(4.13)(30) = 61.95 in 2 Click Click Click The base is a regular pentagon. Finally, calculate your Surface Area. SA = 2B + PH Click SA = 2(61.95) + (30)(10) Click SA = 123.9 + 300 = 423.9 in 2 Click Slide 121 (Answer) / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. Next, calculate the perimeter of your base. Questions to help address MP standards: 6 ft What information are you given? (MP1) Math Practice P = 5(6) = 30 in What is the problem asking? (MP1) Click Use this to find the Lateral Area How can you represent the problem with symbols and numbers? (MP2) LA = PH = 30(10) = 300 in 2 What tools do you need? (MP5) 10 ft Click Can you do this mentally? (MP5) What labels could you use? (MP6) Then, calculate the area of your base, B B = (1/2)aP = (1/2)(4.13)(30) = 61.95 in 2 Click Click Click The base is a regular [This object is a pull tab] pentagon. Finally, calculate your Surface Area. SA = 2B + PH Click SA = 2(61.95) + (30)(10) Click SA = 123.9 + 300 = 423.9 in 2 Click Slide 122 / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. 3 8 6 7 5 Angles are right angles. Then, calculate the area of your base, B First, calculate the perimeter of B = 7(5)+3(3) = 44 units 2 your base. P = 8 + 7 + 5 + 4 + 3 + 3 Finally, calculate your Surface Area. P = 30 units SA = 2B + PH Use this to find the Lateral Area SA = 2(44) + (30)(6) LA = PH = 30(6) = 180 units 2 SA = 88 + 180 = 268 units 2

  51. Slide 122 (Answer) / 311 Other Prisms Example: Find the lateral area and surface area of the right prism. 3 8 Questions to help address MP standards: What information are you given? (MP1) Math Practice What is the problem asking? (MP1) How can you represent the problem with 6 symbols and numbers? (MP2) What tools do you need? (MP5) Can you do this mentally? (MP5) What labels could you use? (MP6) 7 5 Angles are right angles. [This object is a pull tab] Then, calculate the area of your base, B First, calculate the perimeter of B = 7(5)+3(3) = 44 units 2 your base. P = 8 + 7 + 5 + 4 + 3 + 3 Finally, calculate your Surface Area. P = 30 units SA = 2B + PH Use this to find the Lateral Area SA = 2(44) + (30)(6) LA = PH = 30(6) = 180 units 2 SA = 88 + 180 = 268 units 2 Slide 123 / 311 48 Find the lateral area of the right prism. 8 11 The base is a regular hexagon. Slide 123 (Answer) / 311 48 Find the lateral area of the right prism. 8 P = 6(8) = 48 11 Answer LA = 48(11) LA = 528 sq units The base is a [This object is a pull tab] regular hexagon.

  52. Slide 124 / 311 49 Find the total surface area of the right prism. 8 11 The base is a regular hexagon. Slide 124 (Answer) / 311 49 Find the total surface area of the right prism. 8 need to start by finding the apothem = 4√3...30-60-90 triangle Answer P = 6(8) = 48 B = (1/2)(4√3)(48) = 96√3 11 SA = 2(96√3) + 48(11) SA = 192√3 + 528 sq units SA = 860.55 sq units [This object is a pull tab] The base is a regular hexagon. Slide 125 / 311 50 Find the total surface area of the right prism. 10 4 2 3 4 9 All angles are right angles.

  53. Slide 125 (Answer) / 311 50 Find the total surface area of the right prism. 10 4 2 P = 4 + 2 + 2 + 2 + 4 + 3 + 10 + 3 P = 30 3 4 Answer B = 4(3) + 4(3) + 2(1) = 26 SA = 2(26) + 30(9) 9 SA = 52 + 270 SA = 322 sq units [This object is a pull tab] All angles are right angles. Slide 126 / 311 51 The right triangular prism has a surface area of 150 sq ft. Find the height of the prism. 6 A 5 ft 5 B 6 ft C 7.81 ft D 6.38 ft y Slide 126 (Answer) / 311 51 The right triangular prism has a surface area of 150 sq ft. Find the height of the prism. 6 A 5 ft D 5 B 6 ft Need hyp. of the right triangle. C 7.81 ft 5 2 + 6 2 = c 2 c = 7.81 D 6.38 ft Answer y B = (1/2)(5)(6) = 15 P = 5 + 6 + 7.81 = 18.81 150 = 2(15) + 18.81(y) 150 = 30 + 18.81y 120 = 18.81y y = 6.38 ft [This object is a pull tab]

  54. Slide 127 / 311 Surface Area of a Cylinder Return to Table of Contents Slide 128 / 311 Cylinders height base radius height base base base radius A Cylinder is a solid w/ 2 circular bases that lie in || planes. Because each base is a circle, it contains a radius. The remaining measurement that connects the 2 bases is the height of the cylinder. Slide 129 / 311 Cylinders The net of a right cylinder is two circles and a rectangle that forms the lateral surface. radius 8 8 x radius What is the length of x? - The circumference of the circle (base) Click to reveal

  55. Slide 130 / 311 Finding the Surface Area of a Right Cylinder Surface Area : S.A. = 2B + PH B = Area of the circular base = πr 2 C = Perimeter of the Circular base (Circumference) = 2πr H = Height of the prism Lateral Area = CH = 2πrH Base Base height height Lateral Surface Base Base Slide 130 (Answer) / 311 Finding the Surface Area of a Right Cylinder Surface Area : S.A. = 2B + PH B = Area of the circular base = πr 2 The PARCC Reference sheet for C = Perimeter of the Circular base (Circumference) = 2πr the HS level does NOT contain any H = Height of the prism formulas to calculate the Surface Teacher's Note Area (reference sheet is linked to Lateral Area = CH = 2πrH this pull tab - click on tab). Encourage students to either Base memorize the formulas, or draw the Base net each time so that they can break down the solid into smaller 2- D shapes. height height Lateral Surface [This object is a pull tab] Base Base Slide 131 / 311 Finding the Surface Area of a Right Cylinder The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the cylinder between the circular bases. Base Base height height Lateral Surface Base Base Therefore, the Surface Area of a Cylinder can be simplified to the equation below. SA = 2πr 2 + 2πrH

  56. Slide 132 / 311 Finding the Surface Area of a Right Cylinder Example: Find the lateral area and surface area of the right cylinder. LA = 2πrh LA = 2π(4)(8) LA = 64π units 2 8 LA = 201.06 units 2 r = 4 SA = 2πr 2 + 2πrh SA = 2π(4) 2 + 2π(4)(8) SA = 32π + 64π SA = 96π units 2 SA = 301.59 units 2 Slide 132 (Answer) / 311 Finding the Surface Area of a Right Questions to help address MP standards: Math Practice What information are you given? (MP1) Cylinder What is the problem asking? (MP1) How can you represent the problem with Example: Find the lateral area and surface area of the right symbols and numbers? (MP2) cylinder. What tools do you need? (MP5) What labels could you use? (MP6) LA = 2πrh LA = 2π(4)(8) LA = 64π units 2 8 [This object is a pull tab] LA = 201.06 units 2 r = 4 SA = 2πr 2 + 2πrh SA = 2π(4) 2 + 2π(4)(8) SA = 32π + 64π SA = 96π units 2 SA = 301.59 units 2 Slide 133 / 311 Cylinders Example: Find the lateral area and surface area of the right cylinder. 16 2 + h 2 = 34 2 click 256 + h 2 = 1156 click h 2 = 900 34 click h = 30 click Note: 16-30-34 = 2(8-15-17) Pyth. Triple click d = 16 SA = 2πr 2 + 2πrh LA = 2πrh click click SA = 2π(8) 2 + 2π(8)(30) LA = 2π(8)(30) click click SA = 128π + 480π LA = 480π units 2 click click SA = 608π units 2 LA = 1507.96 units 2 click click SA = 1,910.09 units 2 click

  57. Slide 133 (Answer) / 311 Cylinders Questions to help address MP standards: Example: Find the lateral area and surface area of the right What information are you given? (MP1) cylinder. What is the problem asking? (MP1) Math Practice How can you represent the problem with 16 2 + h 2 = 34 2 symbols and numbers? (MP2) click What tools do you need? (MP5) 256 + h 2 = 1156 Can you do this mentally? (MP5) click - Referring to the Pythagorean Triple h 2 = 900 34 16-30-34 = 2(8-15-17) click h = 30 What labels could you use? (MP6) click Note: 16-30-34 = 2(8-15-17) Pyth. Triple click d = 16 [This object is a pull tab] SA = 2πr 2 + 2πrh LA = 2πrh click click SA = 2π(8) 2 + 2π(8)(30) LA = 2π(8)(30) click click SA = 128π + 480π LA = 480π units 2 click click SA = 608π units 2 LA = 1507.96 units 2 click click SA = 1,910.09 units 2 click Slide 134 / 311 Cylinders Example: Find the lateral area and surface area of the right cylinder when the base circumference is 16π ft & the height is 10 ft. C = 2πr click 16π = 2πr 2π 2π click click 8 ft = r click LA = 2πrh SA = 2πr 2 + 2πrh click click LA = 2π(8)(10) SA = 2π(8) 2 + 2π(8)(10) click click LA = 160π ft 2 SA = 128π + 160π click click LA = 502.64 ft 2 SA = 288π ft 2 click click SA = 904.78 ft 2 click Slide 134 (Answer) / 311 Cylinders Example: Find the lateral area and surface area Questions to help address MP standards: of the right cylinder when the base What information are you given? (MP1) Math Practice circumference is 16π ft & the height is 10 ft. What is the problem asking? (MP1) How can you represent the problem with symbols and numbers? (MP2) C = 2πr Would it help to draw a picture? (MP4 & MP5) click 16π = 2πr What tools do you need? (MP5) Can you do this mentally? (MP5) click 2π 2π What labels could you use? (MP6) click 8 ft = r click LA = 2πrh [This object is a pull tab] SA = 2πr 2 + 2πrh click click LA = 2π(8)(10) SA = 2π(8) 2 + 2π(8)(10) click click LA = 160π ft 2 SA = 128π + 160π click click LA = 502.64 ft 2 SA = 288π ft 2 click click SA = 904.78 ft 2 click

  58. Slide 135 / 311 52 Find the lateral area of the right cylinder. h = 12 r = 7 Slide 135 (Answer) / 311 52 Find the lateral area of the right cylinder. LA = 2π(7)(12) Answer h = 12 LA = 168π sq units LA = 527.79 sq units r = 7 [This object is a pull tab] Slide 136 / 311 53 Find the surface area of the right cylinder. Use 3.14 as your value of π & round to two decimal places. A 1200 sq in. B 307.72 sq in. C 835.24 sq in. h = 12 D 1670.48 sq in. r = 7

  59. Slide 136 (Answer) / 311 53 Find the surface area of the right cylinder. Use 3.14 as your value of π & round to two decimal places. C A 1200 sq in. SA = 2π(7)(12) + 2π(7) 2 B 307.72 sq in. Answer SA = 168π + 98π C 835.24 sq in. h = 12 SA = 266π sq in D 1670.48 sq in. SA = 835.24 sq in r = 7 [This object is a pull tab] Slide 137 / 311 54 Find the lateral area of the right cylinder. 13 r = 5 Slide 137 (Answer) / 311 54 Find the lateral area of the right cylinder. 5 2 + h 2 = 13 2 25 + h 2 = 169 Answer h 2 = 144 h = 12 units LA = 2π(5)(12) 13 LA = 120π sq units LA = 376.99 sq units r = 5 [This object is a pull tab]

  60. Slide 138 / 311 55 Find the lateral area of the right cylinder. h = 12 Base area is 36π units 2 Slide 138 (Answer) / 311 55 Find the lateral area of the right cylinder. A = πr 2 36π = πr 2 π π h = 12 Answer 36 = r 2 6 = r LA = 2π(6)(12) LA = 144π sq units Base area is 36π units 2 LA = 452.39 sq units [This object is a pull tab] Slide 139 / 311 56 Find the surface area of the right cylinder. h = 12 Base area is 36π units 2

  61. Slide 139 (Answer) / 311 56 Find the surface area of the right cylinder. A = πr 2 36π = πr 2 π π 36 = r 2 Answer h = 12 6 = r SA = 2π(6)(12) + 2π6 2 SA = 144π + 72 SA = 216π sq units Base area is 36π units 2 SA = 678.58 sq units [This object is a pull tab] Slide 140 / 311 57 The surface area of the right cylinder is 653.12 sq in. Find the height of the cylinder. Use 3.14 as your value of π. h A 7 in. B 8 in. r = 8 in. C 5 in. D 6 in. Slide 140 (Answer) / 311 57 The surface area of the right cylinder is 653.12 sq in. Find the height of the cylinder. Use 3.14 as your value of π. h A 7 in. Answer B 8 in. C r = 8 in. C 5 in. D 6 in. [This object is a pull tab]

  62. Slide 141 / 311 58 A food company packages soup in aluminum cans that have a diameter of 2 1 / 2 inches and a height of 4 inches. Before shipping the cans off to the stores, they add their company label to the can which does not cover the top and bottom. If the company is shipping 200 cans of soup to one store, how much paper material is required to make the labels? Slide 141 (Answer) / 311 58 A food company packages soup in aluminum cans that have a diameter of 2 1 / 2 inches and a height of 4 inches. Before shipping the cans off to the stores, they add their company label to the can which does not cover the top and bottom. If the company is shipping 200 cans of soup to one store, how much paper material is required to 1 can: make the labels? LA = 2πrh LA = 2π(1.25)(4) LA = 10π sq. inches Answer LA = 31.42 sq. inches 200 cans: LA = 200(10π) LA = 2000π sq. inches LA = 6,283.19 sq. inches [This object is a pull tab] Slide 142 / 311 59 Maria's mom baked a cake for her daughter's birthday party. The diameter of the cake is 9 inches and the height is 2 inches. How much base frosting (pink in the picture below) was required to cover the cake?

  63. Slide 142 (Answer) / 311 59 Maria's mom baked a cake for her daughter's birthday party. The diameter of the cake is 9 inches and the height is 2 inches. How much base frosting (pink in the picture below) was required to cover the cake? Area of frosting = LA + area of the top Answer = π(4.5) 2 + 2π(4.5)(2) = 20.25π + 18π = 38.25π sq. inches = 120.17 sq. inches [This object is a pull tab] Slide 143 / 311 Surface Area of a Pyramid Return to Table of Contents Slide 144 / 311 Pyramids A Pyramid is a polyhedron in which the base is a polygon & the lateral faces are triangles with a common vertex. Lateral Edges are the intersection of 2 lateral faces Vertex Lateral Edge Lateral Face Base

  64. Slide 145 / 311 Net This is a right square pyramid. Another name for it is pentahedron. Hedron is a suffix that means face. Why is this a pentahedron? Slide 145 (Answer) / 311 Net There are 5 polygonal faces that create this solid. Answer - "Penta" means 5 This is a right square pyramid. - "Hedon" means face Another name for it is pentahedron. Therefore, it's a solid w/ 5 faces Hedron is a suffix that means face. Why is this a pentahedron? [This object is a pull tab] Slide 146 / 311 Surface Area = Sum of the Areas of all the sides Height of the Triangle Slant Height ℓ The Pyramid has a square base and 4 triangular faces The triangular faces are all isosceles triangles if its a right pyramid. The Height of each triangular face is the Slant Height of the pyramid if it ℓ is a regular pyramid (labeled as , or a cursive lower case L).

  65. Slide 147 / 311 Segment Lengths in a Pyramid Pyramid's Height (h) ℓ Slant Height ( ) Square Base (B) Slide 148 / 311 Segment Lengths in a Pyramid Example: Find the value of x. a 2 + 12 2 = 13 2 a 2 + 144 = 169 a 2 = 25 13 a = 5 12 Note: 5-12-13 Right Triangle Therefore x = 2(5) = 10 x Slide 148 (Answer) / 311 Questions to help address MP standards: What information are you given? (MP1) Segment Lengths in a Pyramid What is the problem asking? (MP1) Math Practice How can you represent the problem with Example: Find the value of x. symbols and numbers? (MP2) What tools do you need? (MP5) Can you do this mentally? (MP5) - Referring to the Pythagorean Triple: 5-12-13 Can you find a shortcut to solve this problem? How would your shortcut make the problem a 2 + 12 2 = 13 2 easier? (MP8) - Referring to the Pythagorean Triple a 2 + 144 = 169 a 2 = 25 [This object is a pull tab] 13 a = 5 12 Note: 5-12-13 Right Triangle Therefore x = 2(5) = 10 x

  66. Slide 149 / 311 Segment Lengths in a Pyramid Example: Find the value of x. Square Base has an area of 64, so 64 = y 2 y = 8, so a = 4 of the right x 8 triangle. 4 2 + 8 2 = x 2 16 + 64 = x 2 x 2 = 80 x = 8.94 units Base Area of the right square pyramid is 64 u 2 . Slide 149 (Answer) / 311 Segment Lengths in a Pyramid Math Practice Questions to help address MP standards: What information are you given? (MP1) What is the problem asking? (MP1) Example: Find the value of x. How can you represent the problem with symbols and numbers? (MP2) What tools do you need? (MP5) Square Base has an area of 64, so 64 = y 2 [This object is a pull tab] y = 8, so a = 4 of the right x triangle. 8 4 2 + 8 2 = x 2 16 + 64 = x 2 x 2 = 80 x = 8.94 units Base Area of the right square pyramid is 64 u 2 . Slide 150 / 311 Segment Lengths in a Pyramid Example: Find the length of the slant height. This is a regular hexagonal pyramid. r = 6 lateral edge = 12 r

  67. Slide 151 / 311 Segment Lengths in a Pyramid First, find the height of the pyramid using Pythagorean Theorem. 6 2 + h 2 = 122 12 h click 36 + h 2 = 144 click h 2 = 108 6 click h = 6√3 = 10.39 click r Note: 30-60-90 triangle click Slide 151 (Answer) / 311 Segment Lengths in a Pyramid First, find the height of the pyramid using Pythagorean Theorem. Questions to help address MP standards: 6 2 + h 2 = 122 12 h What information are you given? (MP1) click What is the problem asking? (MP1) 36 + h 2 = 144 Math Practice How can you represent the problem with click h 2 = 108 symbols and numbers? (MP2) 6 What tools do you need? (MP5) click h = 6√3 = 10.39 Can you do this mentally? (MP5) click - Referring to the 30-60-90 triangle r Can you find a shortcut to solve this problem? Note: 30-60-90 triangle How would your shortcut make the problem easier? (MP8) click - Referring to the 30-60-90 triangle [This object is a pull tab] Slide 152 / 311 Segment Lengths in a Pyramid Second, find the apothem of the hexagonal base. 360 = 60° = central 6 click Note: equilateral click 60 = 30° = top of the . 2 r click 3 2 + a 2 = 6 2 click 9 + a 2 = 36 6 6 click a 2 = 27 a click a = 3√3 = 5.20 3 3 click Note: 30-60-90 triangle click

  68. Slide 152 (Answer) / 311 Segment Lengths in a Pyramid Second, find the apothem of the hexagonal base. Questions to help address MP standards: 360 = 60° = central What information are you given? (MP1) 6 What is the problem asking? (MP1) click Math Practice How can you represent the problem with Note: equilateral symbols and numbers? (MP2) click 60 What tools do you need? (MP5) = 30° = top of the . Can you do this mentally? (MP5) 2 r - Referring to the 30-60-90 triangle click Can you find a shortcut to solve this problem? How would your shortcut make the problem 3 2 + a 2 = 6 2 easier? (MP8) click - Referring to the 30-60-90 triangle 9 + a 2 = 36 6 6 click a 2 = 27 a [This object is a pull tab] click a = 3√3 = 5.20 3 3 click Note: 30-60-90 triangle click Slide 153 / 311 Segment Lengths in a Pyramid Last, find the slant height of your pyramid w/ the apothem & height. ℓ (3√3) 2 + (6√3) 2 = 2 click ℓ 27 + 108 = 2 click ℓ 2 = 135 r click ℓ = 3√15 = 11.62 click ℓ h = 6√3 Click a = 3√3 Click Slide 153 (Answer) / 311 Segment Lengths in a Pyramid Last, find the slant height of your pyramid w/ the apothem & height. Questions to help address MP standards: Math Practice What information are you given? (MP1) What is the problem asking? (MP1) ℓ (3√3) 2 + (6√3) 2 = 2 How can you represent the problem with click ℓ symbols and numbers? (MP2) 27 + 108 = 2 What tools do you need? (MP5) click ℓ 2 = 135 r click ℓ = 3√15 = 11.62 click [This object is a pull tab] ℓ h = 6√3 Click a = 3√3 Click

  69. Slide 154 / 311 60 Find the value of the variable. x 6 16 Slide 154 (Answer) / 311 60 Find the value of the variable. 10 6 2 + 8 2 = x 2 Answer 36 + 64 = x 2 100 = x 2 x 10 = x 6 [This object is a pull tab] 16 Slide 155 / 311 61 Find the value of the variable. x 11 12

  70. Slide 155 (Answer) / 311 61 Find the value of the variable. √85 or 9.22 Answer 6 2 + x 2 = 11 2 x 11 36 + x 2 = 121 x 2 = 85 x =√85 = 9.22 12 [This object is a pull tab] Slide 156 / 311 62 Find the value of the variable. 6 x area of the base is 36 u 2 Slide 156 (Answer) / 311 62 Find the value of the variable. 3√5 or 6.71 Note: Area of base = 36, then side length of square Answer is 6. 6 2 + 3 2 = x 2 6 x 36 + 9 = x 2 x 2 = 45 x =3√5 = 6.71 [This object is a pull tab] area of the base is 36 u 2

  71. Slide 157 / 311 63 Find the value of the slant height. Regular Hexagonal Pyramid r r = 8 lateral edge = 17 Slide 157 (Answer) / 311 63 Find the value of the slant height. Regular Hexagonal Pyramid 16.52 8 2 + h 2 = 17 2 h = 15 Answer apothem in hexagon = 4√3 ℓ (4√3) 2 + 15 2 = 2 r = 16.52 ℓ r = 8 [This object is a pull tab] lateral edge = 17 Slide 158 / 311 64 Find the value of the slant height. Regular Hexagonal Pyramid a a = 9 lateral edge = 12

  72. Slide 158 (Answer) / 311 64 Find the value of the slant height. Regular Hexagonal Pyramid 10.82 apothem in hexagon = 9 9 = x√3, so x = 3√3 & r = 6√3 a Answer (6√3) 2 + h 2 = 12 2 h = 6 a = 9 ℓ 6 2 + 9 2 = 2 r r lateral edge = 12 9 ℓ = 10.82 x x [This object is a pull tab] Slide 159 / 311 Surface Area of a Regular Pyramid Pyramid's Height (h) ℓ Slant Height ( ) Square Base (B) ℓ ℓ Surface Area = B + ½P and Lateral Area = ½P ℓ = Slant Height P = Perimeter of Base B = Area of Base Slide 159 (Answer) / 311 Surface Area of a Regular Pyramid Pyramid's Height (h) The PARCC Reference sheet for ℓ Slant Height ( ) the HS level does NOT contain any formulas to calculate the Surface Teacher's Note Area (reference sheet is linked to this pull tab - click on tab). Encourage students to either memorize the formulas, or draw the Square net each time so that they can Base (B) break down the solid into smaller 2- D shapes. ℓ ℓ Surface Area = B + ½P and Lateral Area = ½P [This object is a pull tab] ℓ = Slant Height P = Perimeter of Base B = Area of Base

  73. Slide 160 / 311 Surface Area of a Regular Pyramid 1 ℓ Why is the Surface Area SA = B + P ? 2 Pyramid's Height (h) Slant Height ( ) ℓ Square Base (B) Surface Area is the sum of all of the areas that make up the solid. In our diagram, these are 4 triangles & 1 square. A square = s s = s 2 = B 1 ℓ A ∆ = s 2 Slide 161 / 311 Surface Area of a Regular Pyramid 1 ℓ Why is the Surface Area SA = B + P ? 2 s ℓ Net of Pyramid Since there are 4 ∆s, we can multiply the area of each ∆ by 4. Therefore, our Surface Area for the Pyramid above is ℓℓ SA = s 2 + 4( 1 / 2 )s SA = s 2 + ( 1 / 2 )(4s) ℓ SA = B + 1 / 2 P Slide 162 / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. LA = 1 / 2 P ℓ LA = 1 / 2 (24)(7) ℓ = 7 LA = 12(7) LA = 84 units 2 SA = B + 1 / 2 P ℓ SA = 6 2 + 1 / 2 (24)(7) SA = 36 + 84 SA = 120 units 2 s = 6

  74. Slide 162 (Answer) / 311 Surface Area of a Regular Pyramid Questions to help address MP standards: Example: Find the lateral area and the surface area of the pyramid. What information are you given? (MP1) Math Practice What is the problem asking? (MP1) How can you represent the problem with symbols and numbers? (MP2) What tools do you need? (MP5) What labels could you use? (MP6) LA = 1 / 2 P ℓ LA = 1 / 2 (24)(7) ℓ = 7 LA = 12(7) LA = 84 units 2 [This object is a pull tab] SA = B + 1 / 2 P ℓ SA = 6 2 + 1 / 2 (24)(7) SA = 36 + 84 SA = 120 units 2 s = 6 Slide 163 / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. First, calculate the slant height. 3 2 + 8 2 = ℓ 2 9 + 64 = ℓ 2 73 = ℓ 2 ℓ = 8.54 Next, calculate the LA & SA h = 8 LA = 1 / 2 P ℓ LA = 1 / 2 (24)(8.54) LA = 12(8.54) s = 6 LA = 102.48 units 2 SA = B + 1 / 2 P ℓ SA = 6 2 + 1 / 2 (24)(8.54) SA = 36 + 102.48 SA = 138.48 units 2 Slide 163 (Answer) / 311 Surface Area of a Regular Pyramid Questions to help address MP standards: What information are you given? (MP1) Math Practice Example: Find the lateral area and the surface area of the What is the problem asking? (MP1) pyramid. How can you represent the problem with symbols and numbers? (MP2) First, calculate the slant height. What tools do you need? (MP5) What labels could you use? (MP6) 3 2 + 8 2 = ℓ 2 9 + 64 = ℓ 2 73 = ℓ 2 ℓ = 8.54 Next, calculate the LA & SA [This object is a pull tab] h = 8 LA = 1 / 2 P ℓ LA = 1 / 2 (24)(8.54) LA = 12(8.54) s = 6 LA = 102.48 units 2 SA = B + 1 / 2 P ℓ SA = 6 2 + 1 / 2 (24)(8.54) SA = 36 + 102.48 SA = 138.48 units 2

  75. Slide 164 / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. 10 ℓ 8 e = 10 First, calculate the slant height. 8 2 + ℓ 2 = 10 2 click 64 + ℓ 2 = 100 s = 16 click ℓ 2 = 36 ℓ = 6 click Slide 164 (Answer) / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the Questions to help address MP standards: pyramid. What information are you given? (MP1) What is the problem asking? (MP1) 10 How can you represent the problem with ℓ Math Practice symbols and numbers? (MP2) What tools do you need? (MP5) Can you solve this problem mentally? (MP5) 8 - Referring to Pythagorean Triple: 6-8-10 = e = 10 2(3-4-5) First, calculate the What labels could you use? (MP6) Can you find a shortcut to solve the problem? slant height. How would your shortcut make the problem easier? (MP8) 8 2 + ℓ 2 = 10 2 click [This object is a pull tab] 64 + ℓ 2 = 100 s = 16 click ℓ 2 = 36 ℓ = 6 click Slide 165 / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. Next, calculate the LA & SA LA = 1 / 2 P ℓ click LA = 1 / 2 (64)(6) click e = 10 LA = 32(6) click LA = 192 units 2 click SA = B + 1 / 2 P ℓ click SA = 16 2 + 1 / 2 (64)(6) s = 16 click SA = 256 + 192 click SA = 448 units 2 click

  76. Slide 165 (Answer) / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. Next, calculate the LA & SA Questions to help address MP standards: LA = 1 / 2 P ℓ Math Practice What information are you given? (MP1) What is the problem asking? (MP1) click LA = 1 / 2 (64)(6) How can you represent the problem with click symbols and numbers? (MP2) LA = 32(6) e = 10 What tools do you need? (MP5) click LA = 192 units 2 What labels could you use? (MP6) click SA = B + 1 / 2 P ℓ click SA = 16 2 + 1 / 2 (64)(6) s = 16 [This object is a pull tab] click SA = 256 + 192 click SA = 448 units 2 click Slide 166 / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. Regular Pentagonal Pyramid a a = 4 lateral edge = 8 Slide 167 / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. First, find the radius & side length of the regular pentagon using the apothem & trigonometric ratios 360 = 72° = central 5 Click 72° Click 72 r 36°36° = 36° = top 2 Click Click of the . 4 Click x x tan36 = 4 Click 4 cos36 = r Click x = 4tan36 = 2.91 rcos36 = 4 Click cos36 cos36 Click Therefore, s = 2(2.91) = 5.82 Click r = 4.94 Click

  77. Slide 167 (Answer) / 311 Surface Area of a Regular Pyramid Example: Find the lateral area and the surface area of the pyramid. First, find the radius & side length of the regular pentagon using the Questions to help address MP standards: What information are you given? (MP1) apothem & trigonometric ratios Math Practice What is the problem asking? (MP1) How can you represent the problem with 360 = 72° = central symbols and numbers? (MP2) 5 Click What tools do you need? (MP5) 72° How is right triangle trigonometry used to Click calculate the segment lengths in regular 72 r 36°36° = 36° = top polygons? (MP7) 2 Click Click of the . 4 Click x x tan36 = 4 [This object is a pull tab] Click 4 cos36 = r Click x = 4tan36 = 2.91 rcos36 = 4 Click cos36 cos36 Click Therefore, s = 2(2.91) = 5.82 Click r = 4.94 Click Slide 168 / 311 Surface Area of a Regular Pyramid Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pythagorean Theorem. 2.912 + ℓ 2 = 8 2 click 8.4681 + ℓ 2 = 64 8 ℓ click ℓ 2 = 55.5319 click ℓ = 7.45 click 2.91 Slide 168 (Answer) / 311 Surface Area of a Regular Pyramid Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pythagorean Theorem. Questions to help address MP standards: What information are you given? (MP1) Math Practice 2.912 + ℓ 2 = 8 2 What is the problem asking? (MP1) click How can you represent the problem with 8.4681 + ℓ 2 = 64 8 symbols and numbers? (MP2) ℓ click What tools do you need? (MP5) ℓ 2 = 55.5319 How is Pythagorean Theorem used to calculate click ℓ = 7.45 the segment lengths of pyramids? (MP7) click 2.91 [This object is a pull tab]

  78. Slide 169 / 311 Surface Area of a Regular Pyramid Last, find the lateral area & surface area of the pyramid. SA = B + 1 / 2 P ℓ LA = 1 / 2 P ℓ click click SA = 1 / 2 (4)(29.1) + 1 / 2 (29.1)(7.45) LA = 1 / 2 (29.1)(7.45) click click SA = 58.2 + 108.40 LA = 108.40 units 2 click click SA = 166.6 units 2 click Slide 169 (Answer) / 311 Surface Area of a Regular Pyramid Last, find the lateral area & surface area of the pyramid. Questions to help address MP standards: What information are you given? (MP1) SA = B + 1 / 2 P ℓ Math Practice What is the problem asking? (MP1) LA = 1 / 2 P ℓ click click How can you represent the problem with SA = 1 / 2 (4)(29.1) + 1 / 2 (29.1)(7.45) LA = 1 / 2 (29.1)(7.45) symbols and numbers? (MP2) click click SA = 58.2 + 108.40 What tools do you need? (MP5) LA = 108.40 units 2 click What labels could you use? (MP6) click SA = 166.6 units 2 click [This object is a pull tab] Slide 170 / 311 65 Find the lateral area of the right pyramid. ℓ = 9 s = 10

  79. Slide 170 (Answer) / 311 65 Find the lateral area of the right pyramid. LA = 1 / 2 P ℓ Answer LA = 1 / 2 (40)(9) ℓ = 9 LA = 180 units 2 s = 10 [This object is a pull tab] Slide 171 / 311 66 Find the surface area of the right pyramid. ℓ = 9 s = 10 Slide 171 (Answer) / 311 66 Find the surface area of the right pyramid. ℓ = 9 SA = B + 1 / 2 P ℓ Answer SA = 10 2 + 1 / 2 (40)(9) SA = 100 + 180 SA = 280 units 2 s = 10 [This object is a pull tab]

  80. Slide 172 / 311 67 Find the lateral area of the right pyramid. e = 10 base area = 16 Slide 172 (Answer) / 311 67 Find the lateral area of the right pyramid. B = 16, so s = 4 & small leg of rt ∆ is 2 e = 10 2 2 + ℓ 2 = 10 2 ℓ = 9.80 Answer LA = 1 / 2 P ℓ base LA = 1 / 2 (16)(9.80) area = 16 LA = 78.4 units 2 [This object is a pull tab] Slide 173 / 311 68 Find the surface area of the right pyramid. e = 10 base area = 16

  81. Slide 173 (Answer) / 311 68 Find the surface area of the right pyramid. ℓ = 9.80 (see previous e = 10 slide) Answer SA = B + 1 / 2 P ℓ SA = 16 + 78.4 SA = 94.4 units 2 base area = 16 [This object is a pull tab] Slide 174 / 311 69 Find the lateral area of the right pyramid. a = 5 h = 12 a Regular Octagonal Pyramid Slide 174 (Answer) / 311 69 Find the lateral area of the right pyramid. a = 5 tan22.5 = x h = 12 a = 5 & h = 12 5 5 2 + 12 2 = ℓ 2 x = 2.07 a ℓ = 13 units 1 side = 4.14 Answer P = 8(4.14) = 33.12 LA = 1 / 2 P ℓ LA = 1 / 2 (33.12)(13) LA = 215.28 units 2 [This object is a pull tab] Regular Octagonal Pyramid

  82. Slide 175 / 311 70 Find the surface area of the right pyramid. a = 5 h = 12 a Regular Octagonal Pyramid Slide 175 (Answer) / 311 70 Find the surface area of the right pyramid. a = 5 ℓ = 13 units P = 33.12 units h = 12 B = 1 / 2 aP ℓ a Answer B = 1 / 2 (5)(33.12) B = 82.8 units 2 SA = B + 1 / 2 P ℓ SA = 82.8 + 215.28 SA = 298.08 units 2 [This object is a pull tab] Regular Octagonal Pyramid Slide 176 / 311 71 Find the surface area of the right pyramid. 8 12 30 Hint: The pyramid is NOT regular. Hint So, B + 1/2 P ℓ doesn't work. Instead, draw a net of the pyramid & find each area.

  83. Slide 176 (Answer) / 311 71 Find the surface area of the right pyramid. ℓ 1 = 17 8 ℓ 2 = 10 Answer 12 SA = sum of all areas SA = 30(12) + 2( 1 / 2 )(12)(17) + 30 2( 1 / 2 )(30)(10) Hint: The pyramid is NOT regular. SA = 360 + 204 + 300 Hint So, B + 1/2 P ℓ doesn't work. SA = 864 units 2 [This object is a pull tab] Instead, draw a net of the pyramid & find each area. Slide 177 / 311 Surface Area of a Cone Return to Table of Contents Slide 178 / 311 Cones S Base l a n t H e i g height h t ℓ r Lateral Surface Slant Height The Base of the cone is a circle ℓ The length of the circular portion of the Lateral Surface is the same as the Circumference of the Circlular Base. The Slant Height is the length of the diagonal slant of the cone from the top to the edge of the base. The Height of the cone is the length from the top to the center of the circular base.

  84. Slide 179 / 311 Finding the Surface Area of a Right Cone Surface Area = Area of the Base + Lateral Area Lateral Area= ½P ℓ Base S.A. = B + ½P ℓ ℓ = Slant Height P = Perimeter of Circular Base B = Area of Circular Base Lateral Surface Because the base is a circle. P = Circumference = 2πr Slant Height L.A. = ½(2πr) ℓ = πr ℓ ℓ S.A. = πr 2 + πr ℓ Slide 179 (Answer) / 311 Finding the Surface Area of a Right Cone Surface Area = Area of the Base + Lateral Area The PARCC Reference sheet for Base Lateral Area= ½P ℓ the HS level does NOT contain any S.A. = B + ½P ℓ formulas to calculate the Surface Teacher's Note Area (reference sheet is linked to ℓ = Slant Height this pull tab - click on tab). P = Perimeter of Circular Base Encourage students to either B = Area of Circular Base memorize the formulas, or draw the Lateral Surface net each time so that they can Because the base is a circle. break down the solid into smaller 2- P = Circumference = 2πr Slant Height D shapes. L.A. = ½(2πr) ℓ = πr ℓ ℓ [This object is a pull tab] S.A. = πr 2 + πr ℓ Slide 180 / 311 Cones Example: Find the lateral area and surface area of the right cone. r = 6 LA = πr ℓ click = π(6)(8) click LA = 48π units 2 ℓ = 8 click LA = 150.80 units 2 click SA = πr 2 + πr ℓ click = π(6) 2 + π(6)(8) click = 36π + 48π click SA = 84π units 2 click SA = 263.89 units 2 click

  85. Slide 180 (Answer) / 311 Cones Example: Find the lateral area and surface area of the right cone. r = 6 LA = πr ℓ Questions to help address MP standards: What information are you given? (MP1) click = π(6)(8) Math Practice What is the problem asking? (MP1) click How can you represent the problem with LA = 48π units 2 ℓ symbols and numbers? (MP2) = 8 click LA = 150.80 units 2 What tools do you need? (MP5) What labels could you use? (MP6) click SA = πr 2 + πr ℓ click = π(6) 2 + π(6)(8) click = 36π + 48π [This object is a pull tab] click SA = 84π units 2 click SA = 263.89 units 2 click Slide 181 / 311 Cones Example: Find the lateral area and surface area of the right cone. C = 12π units C = 2πr click 12π = 2πr click 2π 2π h = 8 click 6 units = r click 6 2 + 8 2 = ℓ 2 click 36 + 64 = ℓ 2 click 100 = ℓ 2 click 10 units = ℓ click Slide 181 (Answer) / 311 Cones Example: Find the lateral area and surface area of the right cone. Questions to help address MP standards: What information are you given? (MP1) C = 12π units What is the problem asking? (MP1) C = 2πr How could you start this problem? (MP1) click What does 12π represent in this problem? 12π = 2πr (MP2) click 2π 2π Math Practice h = 8 What other value can it help you calculate? click 6 units = r (MP2) How does Pythagorean Theorem related to click calculating the slant height? (MP7) How can you represent the problem with symbols and numbers? (MP2) Can you find a shortcut to solve the problem? 6 2 + 8 2 = ℓ 2 How would your shortcut make the problem click easier? (MP8) 36 + 64 = ℓ 2 What tools do you need? (MP5) click 100 = ℓ 2 [This object is a pull tab] What labels could you use? (MP6) click 10 units = ℓ click

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