Slide 1 / 108 Slide 2 / 108 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative Volume This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course March 7, 2012 materials to parents, students and others. www.njctl.org Click to go to website: www.njctl.org Slide 3 / 108 Slide 4 / 108 Table of Contents Volume of Prisms Volume of Cylinders Volume of a Prism Volume of Pyramids Volume of Cones Volume of a Sphere Cavalieri's Principle Corresponding Parts of Similar Solids Return to Table of Contents Coordinates in Space Click on the topic to go to that section Slide 5 / 108 Slide 6 / 108 Finding the Volume of a Prism Definition of Volume - Base The amount of cubic units that a solid can hold. V = Bh h height Specific Prisms Box: V = Lwh Cube: V=s 3 Base w Where area used square units, L volume will use cubic units.
Slide 7 / 108 Slide 8 / 108 Does a prism need to be a right prism for the Example: Find the volume of the box with length 2, volume formula to work? width 6, and height 5. Think of a ream of paper If the stack is fanned, it Stacked nicely it still has 500 sheets. has 500 sheets. So the volume doesn't change if the prism, stack of paper, is right or oblique. The formula V=BH works for all prisms Slide 9 / 108 Slide 10 / 108 Example: The volume of a box is 48 ft 3 . If the height Example: Find the volume of the cube with edges is 4ft and width is 6ft, what is the length. of 7. Slide 11 / 108 Slide 12 / 108 Example: Find the volume of a cube is 64 m 3 , what Example: Find the volume of the prism with height is area of one face? 8 and hexagon base with apothem 4.
Slide 13 / 108 Slide 14 / 108 1 What is the volume of a box with edges of 4, 5, and 7? 2 What is the volume of a box with edges of 8, 6, and 10? Slide 15 / 108 Slide 16 / 108 If the volume of a box is 64 u 3 and has height 8 and width 3 What is the volume of a cube with edges of 5 units? 4 4, what is the length? Slide 17 / 108 Slide 18 / 108 If the volume of a box is 120 u 3 and has height 6 and If a cube has volume 27 u 3 , what is the cubes surface 5 6 length 4, what is the width? area?
Slide 19 / 108 Slide 20 / 108 7 Find the volume of the prism. 8 Find the volume of the prism. 15 6 6 20 7 12 6 2 Slide 21 / 108 Slide 22 / 108 Finding the Volume of a Cylinder Volume of a Cylinder r base V = Bh height V = πr 2 h r base Return to Table of Contents Slide 23 / 108 Slide 24 / 108 Example: Find the volume of the cylinder with radius 4 Example: The surface area of a cylinder is 425 u 2 , and height 11. what is the volume?
Slide 25 / 108 Slide 26 / 108 9 Find the volume of the cylinder with radius 6 and 10 Find the volume of the cylinder with circumference 18 π units and height 6? height 8. Slide 27 / 108 Slide 28 / 108 The volume of a cylinder is 108 u 3 , what is the 11 Find the volume of the cylinder with a surface area 12 108 u 2 . surface area? Slide 29 / 108 Slide 30 / 108 13 The height of a cylinder doubles, what happens to the 14 The radius of a cylinder doubles, what happens to the volume? volume? A Doubles A Doubles Quadruples Quadruples B B Depends on the cylinder Depends on the cylinder C C Cannot be determined Cannot be determined D D
Slide 31 / 108 Slide 32 / 108 15 A 3" hole is drilled through a solid cylinder with a diameter of 4" forming a tube. What is the volume of the tube? 4" Volume of a Pyramid 3" 24" Return to Table of Contents Slide 33 / 108 Slide 34 / 108 Example: Find the volume of the pyramid. Finding the Volume of a Pyramid 6 Pyramid's Height (h) V = 1/3 Bh 4 Slant Height ( l ) 5 Square Base (B) Slide 35 / 108 Slide 36 / 108 Example: Find the volume of the pyramid. Example: Find the volume of the pyramid. 6 5 4 8 5 8
Slide 37 / 108 Slide 38 / 108 16 Find the volume of the pyramid. 17 Find the volume of the pyramid. 5 8 6 6 7 6 Slide 39 / 108 Slide 40 / 108 18 Find the volume of the pyramid. A truncated pyramid is a pyramid with its top cutoff parallel to its base. 10 Find the volume of the truncated pyramid shown. 12 3 2 2 12 9 6 6 Slide 41 / 108 Slide 42 / 108 19 Find the volume of the pyramid. Volume of a Cone 3 2 2 12 8 8 Return to Table of Contents
Slide 43 / 108 Slide 44 / 108 Example: Find the volume of the cone. Finding the Volume of a Cone r= 7 V = 1/3 Bh 9 S l a n t H V = 1/3 π r 2 h e i g height h t l r Slide 45 / 108 Slide 46 / 108 Example: Find the volume of the cone, with lateral area 15 π Example: Find the volume of the cone. units 2 and slant height 5. 12 r= 4 Slide 47 / 108 Slide 48 / 108 20 What is the volume of the cone? 21 What is the volume of the cone? 10 40 o 8 d=10
Slide 49 / 108 Slide 50 / 108 22 What is the volume of the truncated cone? 6 r=4 Volume of a Sphere 6 r=8 Return to Table of Contents Slide 51 / 108 Slide 52 / 108 Example: Find the volume of the sphere. Finding the Volume of a Sphere 9 r V = 4/3 π r 3 Slide 53 / 108 Slide 54 / 108 Example: Find the volume of the sphere. Example: Find the volume of the sphere. Great Circle: A=25 π u 2 Surface Area: SA=36 π u 2
Slide 55 / 108 Slide 56 / 108 23 Find the volume of the sphere. 24 Find the volume of the sphere. 4 6 Slide 57 / 108 Slide 58 / 108 25 Find the volume of the sphere. 26 Find the volume of the sphere. Great Circle: A= 16 π u 2 Surface Area: SA= 16 π u 2 Slide 59 / 108 Slide 60 / 108 27 Find the volume of the sphere. 3 Cavalieri's Principle Cross Section: A= 16 π u 2 Return to Table of Contents
Slide 61 / 108 Slide 62 / 108 Cavalieri's Principle 14 14 14 If two solids are the same height, and the area of their cross sections are equal, then the two solids will have the same volume Which solid has the greatest volume? Slide 63 / 108 Slide 64 / 108 A sphere is submerged in a cylinder. What is the 28 These 2 surfaces have the same volume, find x. volume of the cylinder not occupied by the sphere? volume of cylinder - volume of sphere 11 11 The result shows that the left over volume is equal to 2 of what other solid? According to Cavalieri, what can be said about the cross section? Slide 65 / 108 Slide 66 / 108 29 These 2 surfaces have the same volume, find x. Corresponding Parts of Similar 12 12 Solids Return to Table of Contents
Slide 67 / 108 Slide 68 / 108 30 The pyramid on the left is the preimage and is similar Corresponding parts of similar figures are similar. to the image on the right. Find the value of x. The prisms shown are similar. Find x and y. 16 h y 4 6 2 y 3 x 8 2 9 The ratio of similitude, k, is the common value that is multiplied to preimage to get to the image. x 8 If the smaller prism is the preimage, what is k? If the larger prism is the preimage, what is k? Slide 69 / 108 Slide 70 / 108 31 The pyramid on the left is the preimage and is similar 32 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of y. to the image on the right. Find the value of h. 16 16 y y h h 3 3 8 8 2 2 x x 8 8 Slide 71 / 108 Slide 72 / 108 Consider the example of the prisms from earlier. The Comparing Similar Figures ratio of similitude from the smaller surface to the larger is 3/2. length in image = k length in preimage area in image 4 6 = k 2 2 3 area in preimage 6 9 volume in image = k 3 Look at the area of their bases: how do they compare? volume in preimage How do their volumes compare?
Slide 73 / 108 Slide 74 / 108 Example: 33 The scale factor of 2 similar pyramids is 4. If the area of the base of the larger one is 64 u 2 , what is area of the smaller one? r = 9 r = 3 How many times bigger is the great circle on the right? How many times bigger is the surface area on the right? How many times bigger is the voklume on the right? Slide 75 / 108 Slide 76 / 108 34 The scale factor of 2 similar right square pyramids is 3. 35 An architect builds a scale model of a home using 2 in to If the area of the base of the larger one is 36 u 2 and its 5 ft. scale. Given the view of the roof of the model, how height is 12, what is lateral area of the smaller one? much roofing material is needed for the house? 12in 6in 8in 5in 4in 3in Slide 77 / 108 Slide 78 / 108 Graphing in space requires the x-, y-, and z-axes. Coordinates in Space z - + Graphing Distance - y + Midpoint Diagonal of a Box x Equation of a Sphere + - Return to Table of Contents
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