Slide 1 / 311 Slide 2 / 311 Geometry 3D Geometry 2015-10-28 www.njctl.org Slide 3 / 311 Slide 4 / 311 Table of Contents Throughout this unit, the Standards for Mathematical Practice are used. Intro to 3-D Solids Click on the topic to go to that section Views & Drawings of 3-D Solids MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. Surface Area of a Prism MP3: Construct viable arguments and critique the reasoning of Surface Area of a Cylinder others. MP4: Model with mathematics. Surface Area of a Pyramid MP5: Use appropriate tools strategically. Surface Area of a Cone MP6: Attend to precision. MP7: Look for & make use of structure. Volume of a Prism MP8: Look for & express regularity in repeated reasoning. Volume of a Cylinder Additional questions are included on the slides using the "Math Volume of a Pyramid Practice" Pull-tabs (e.g. a blank one is shown to the right on Volume of a Cone this slide) with a reference to the standards used. Surface Area & Volume of Spheres If questions already exist on a slide, then the specific MPs that Cavaleri's Principle the questions address are listed in the Pull-tab. Similar Solids PARCC Sample Questions Slide 5 / 311 Slide 6 / 311 Intro to 3-D Solids 2-dimensional drawings use only the x and y axes X Y Length Intro to 3-Dimensional Solids h t d w i Y width X Length w i d t h Length X Return to Table of Contents Y
Slide 7 / 311 Slide 8 / 311 Intro to 3-D Solids Intro to 3-D Solids 3-dimensional drawings include the x, y and z-axis. Z Y The z-axis is the third dimension. x height The third dimension is the height of the figure Z height X height Y height X Y X X Y Y Slide 9 / 311 Slide 10 / 311 Intro to 3-D Solids Intro to 3-D Solids Y To give a figure more of a 3-dimensional look, lines that are not visible from the angle the figure is being viewed Z are drawn as dashed line segments. These are called r hidden lines. X Z height X height Y height X X Y Y Slide 11 / 311 Slide 12 / 311 Intro to 3-D Solids Intro to 3-D Solids A Polyhedron (pl. Polyhedra) is a solid that is bounded by The 3-Dimensional Figures discussed in polygons, called faces. An edge is the line segment formed by the this unit are: intersection of 2 faces. A vertex is a point where 3 or more edges meet Cylinders Prisms Vertex Edge Pyramids Face
Slide 13 / 311 Slide 14 / 311 Intro to 3-D Solids Right Vs. Oblique The 3-Dimensional Figures discussed in this unit are: In Right Prisms & Cylinders, the bases are aligned directly above one another. The edges are perpendicular with both bases. Cones: Spheres: . C Right Right Slide 15 / 311 Slide 16 / 311 Right Vs. Oblique Right Vs. Oblique In Right Pyramids & Cones, the vertex is aligned directly In Oblique Prisms & Cylinders, the bases are not aligned above the center of the base. directly above one another. The edges are not perpendicular with the bases. In Oblique Pyramids & Cones, the vertex is not aligned directly above the center of the base. Oblique Right Right Oblique Slide 17 / 311 Slide 18 / 311 Intro to 3-D Solids Intro to 3-D Solids Prisms have 2 congruent polygonal bases. The polygons that make up the surface of the figure are The sides of a base are called base edges. called faces. The bases are a type of face and are The segments connecting corresponding vertices are lateral edges. parallel and congruent to each other. The lateral A B edges are the sides of the lateral faces. A C In this diagram: B There are 2 bases: ABC & XYZ. X Y C There are 3 lateral faces: AXBY, BYCZ, & CZAX. In this diagram: Z X Y This prism has a total of 5 faces. There are 6 vertices: A, B, C, X, Y, & Z There are 2 bases: ABC & XYZ. Z There are 6 base edges: AB, BC, AC, XY, YZ, & XZ. There are 3 lateral edges: AX, BY, & CZ. This prism has a total of 9 edges.
Slide 19 / 311 Slide 20 / 311 Choose all of the lateral edges. 2 1 Choose all of the base edges. AB A AB B C A A CD B D B C B DE A F E D C ER FS C F E N BN P D CP D N Q P M DQ E FA E S R Q M QR F CD R F S MS G G NP H AM BC H I CP DQ I Slide 21 / 311 Slide 22 / 311 3 Chooses all of the bases. 4 Chooses all of the lateral faces. B C B C A A AFSM A AFSM A D D F E FERS FERS B B F E C EDQR C EDQR N P N P ABCDEF ABCDEF D D Q M Q M R S R CDQP S CDQP E E F BCPN F BCPN G MNPQRS G MNPQRS H ABNM H ABNM Slide 23 / 311 Slide 24 / 311 Intro to 3-D Solids 5 Chooses all of the faces. B C A pyramid has 1 base with vertices and the lateral A D edges go to a single vertex. AFSM A F E A FERS B N P C EDQR This pyramid has: Q 6 lateral edges, M D ABCDEF S R 6 base edges, 12 edges (total) CDQP E 7 vertices P N BCPN F M Q MNPQRS G S R H ABNM
Slide 25 / 311 Slide 26 / 311 Intro to 3-D Solids 6 Choose all of the base edges. V A pyramid has faces that are polygons: VN A 1 base and triangles that are the lateral faces. KN B A VL C N This pyramid has: K M D LM 6 lateral faces, 1 base, VM E L 7 faces (total) N P VK F M Q G KL S R H NM Slide 27 / 311 Slide 28 / 311 7 Choose all of the lateral edges. 8 How many edges does the pyramid have? V V A VN KN B VL N C K M N LM K M D L VM E L F VK G KL H NM Slide 29 / 311 Slide 30 / 311 10 Choose all of the bases. 9 Choose all of the lateral faces. V V KNV KNV A A NMV NMV B B N N C KLMN C KLMN K M K M VML VML D D L L KLV KLV E E
Slide 31 / 311 Slide 32 / 311 Intro to 3-D Solids 11 How many faces does the pyramid have? A cylinder has 2 bases which are congruent circles. The V lateral face is a rectangle wrapped around the circles. A . A & B N K M are the bases of the cylinder. . L B A cylinder can also be formed by rotating a rectangle about an axis. Click for sample animation Slide 33 / 311 Slide 34 / 311 Intro to 3-D Solids Intro to 3-D Solids A sphere is a 3-dimensional circle in that every point on the sphere is A cone, like a pyramid, has one base which is a circle. the same distance from the center. . N N is the base of the cone. . C V V is the vertex of the cone. A cone can also be formed by rotating a right triangle about Similar to a circle, a sphere is named by its center one of its legs. point. Sphere C is the solid shown above. Click for sample animation Slide 35 / 311 Slide 36 / 311 12 Which solids have 2 bases? 13 Which solid has one vertex? Prism Prism A A Pyramid Pyramid B B C Cylinder C Cylinder Cone Cone D D Sphere Sphere E E
Slide 37 / 311 Slide 38 / 311 14 Which solid has more base edges than 15 Which solid(s) have no vertices? lateral edges? Prism Prism A A Pyramid Pyramid B B Cylinder Cylinder C C Cone Cone D D E Sphere E Sphere Slide 39 / 311 Slide 40 / 311 Intro to 3-D Solids 16 Which solid is formed when rotating an isosceles triangle Euler's Theorem states that the number of faces (F), vertices (V), about its altitude? and edges (E) satisfy the formula F + V = E + 2 A a prism A A B B a cylinder C C a pyramid N P X Y D a cone M Q R S Z E a sphere F = 5 F = 7 V = 6 V = 7 E = 9 E = 12 5 + 6 = 9 + 2 7 + 7 = 12 + 2 11 = 11 14 = 14 Slide 41 / 311 Slide 42 / 311 Intro to 3-D Solids Intro to 3-D Solids Example: Example: A solid has 12 faces, 2 decagons and 10 trapezoids. A solid has 9 faces, 1 octagon and 8 triangles. How How many vertices does the solid have? many vertices does the solid have? On their own, the 2 decagons & 10 trapezoids have What information do you have? 2(10) + 10(4) = 60 edges. In a 3-D solid, each side is 9 faces & the 2 types of faces shared by 2 polygons. Therefore, the number of edges click How are the number of edges in the solid is 60/2 = 30. in the 2-D faces, related to the V + F = E + 2 number of edges in the click polyhedron? Write a number What is the problem asking? V + 12 = 30 + 2 sentence to describe this Create an equation to represent click V + 12 = 32 situation. the problem. click V = 20 V + F = E + 2 (1(8) + 8(3))/2 click 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 Slide 44 / 311 17 A solid has 10 faces, one of them being a nonagon 18 A solid has 12 faces, all of them being pentagons. and 9 triangles. How many vertices does it have? How many vertices does it have? 8 30 A A 9 20 B B 10 15 C C D 18 D 10 Slide 45 / 311 Slide 46 / 311 Intro to 3-D Solids 19 A solid has 8 faces, all of them being triangles. How many vertices does it have? Cross-Section A 24 A cross-section is the locus of points of the intersection of a plane and a 3-D solid. 12 B 8 C 6 D Slide 47 / 311 Slide 48 / 311 Cross-Section Cross-Section Think about it as if the plane were a knife and you were Cross-sections of a surface are a 2-dimensional figure. cutting the shape, what would the cut look like? Cross-sections of a solid are a 2-dimensional figure and its interior. Circle Ellipse The top can be removed to see the cross section. (Try it out) Parabola (with the inner section shaded)
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