Geometry Volume of Prisms and Cylinders CONFIDENTIAL

Warm Up 1) Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled? 2) Max can type 40 words per minute. He estimates that his term paper contains about 5000 words, and he takes a 15-minute break for every 45 minutes of typing. About how much time will it take Max to type his term paper? CONFIDENTIAL

Volume of Prisms and Cylinders The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Next Page: CONFIDENTIAL

Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume CONFIDENTIAL

The volume of a prism with base area B and height h is V = Bh. Next Page: CONFIDENTIAL

The volume of a cube with edge length s is V = s . The volume of a right rectangular prism with length l, width w, and height h is V = lwh. The volume of a cube with edge length s is V = s . 3 CONFIDENTIAL

Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. 10 cm 12 cm 8 cm A). volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh. V = lwh = (10)(12)(8) = 980 cm 3 Next Page: CONFIDENTIAL

B). A cube with edge length 10 cm V= s Volume of a cube = 10 = 1000 cm Substitute 10 for s. 3 Next Page: CONFIDENTIAL

C). A right regular pentagonal Prism with base edge length 5 m and height 7 m. 36° 7 m 5 m Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36° Step:1 tan 36° = 2.5/a The leg of the triangle is half the side length. Or 2.5 m. a = 2.5/tan 36° Solve for a. Next Page: CONFIDENTIAL

Use the value of a to find the base area. 36° 7 m 5 m Step:2 Use the value of a to find the base area. Step:3 Use the base area to find the volume. CONFIDENTIAL

Now you try! 1) Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long. CONFIDENTIAL

Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon = 0.134 ft ) 3 8 ft 120 ft 60 ft Next Page: CONFIDENTIAL

Find the volume of the aquarium in cubic feet. Step:1 Find the volume of the aquarium in cubic feet. V = lwh = (120)(60)(8) = 57,600 cm 3 8 ft 120 ft 60 ft Next Page: CONFIDENTIAL

8 ft 120 ft 60 ft Step:2 Next Page: CONFIDENTIAL

8 ft 120 ft 60 ft Step:3 The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds CONFIDENTIAL

Now you try! 2) Estimate the volume in gallons and the weight of the water in the aquarium below if the height were doubled. 8 ft 120 ft 60 ft CONFIDENTIAL

Cavalieri’s principle also relates to cylinders Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. CONFIDENTIAL

The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h. 2 r h CONFIDENTIAL

Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. 12 cm 8 cm A). Next Page: CONFIDENTIAL

Use the base area to find the radius. A cylinder with a base area of 36 in and a height equal to twice the radius. 2 Step:1 Use the base area to find the radius. 12 cm 8 cm Use the radius to find the height. The height is equal to twice the radius. Step:2 Step:3 Use the radius and height to find the volume. CONFIDENTIAL

3). Find the volume of a cylinder with a diameter of 16in. and Now you try! 3). Find the volume of a cylinder with a diameter of 16in. and a height of 17 in. Give your answer both in terms of and rounded to the nearest tenth. CONFIDENTIAL

Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume. 6 m 12 m CONFIDENTIAL

4) The length, width, and height of the prism are Now you try! 4) The length, width, and height of the prism are doubled. Describe the effect on the volume. 1.5 ft 4 ft 3 ft CONFIDENTIAL

Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. 5 m 8 m 9 m 6 m CONFIDENTIAL

5) Find the volume of the composite figure. Round to Now you try! 5) Find the volume of the composite figure. Round to the nearest tenth. 3 cm 5 cm CONFIDENTIAL

Now some problems for you to practice ! CONFIDENTIAL

1. Find the volume of each prism. Assessment 1. Find the volume of each prism. A) B) 6 m 8 m 6 cm 4 cm 9 cm CONFIDENTIAL

(Hint: 1 gallon = 0.134 cubic feet) 2. The world’s largest ice cream cake, built in New York City on may 25, 2004, was approximately a 19 ft by 9 ft by 2 ft rectangular prism. Estimate the volume of the ice cream cake in gallons. If the density of the ice cream cake was 4.73 pounds per gallon, estimate the weight of the cake. (Hint: 1 gallon = 0.134 cubic feet) 2 ft 19 ft 9 ft CONFIDENTIAL

3. Find the volume of each cylinder 3. Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. A) B) 12 ft 10 ft 5 m 3 m CONFIDENTIAL

A) The dimensions are multiplied by ¼ . B) The dimensions are tripled. 4. Describe the effect of each change on the volume of the given figure. A) The dimensions are multiplied by ¼ . B) The dimensions are tripled. 7 in. 2 in. 8 ft 4 ft 12 ft CONFIDENTIAL

5. Find the volume of each composite figure. Round to the nearest tenth. B) 6 ft 4 ft 12 ft 10 in. 5 in. 15 in. 14 ft CONFIDENTIAL

Let’s review CONFIDENTIAL

Volume of Prisms and Cylinders The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Next Page: CONFIDENTIAL

Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume CONFIDENTIAL

The volume of a prism with base area B and height h is V = Bh. Next Page: CONFIDENTIAL

The volume of a cube with edge length s is V = s . The volume of a right rectangular prism with length l, width w, and height h is V = lwh. The volume of a cube with edge length s is V = s . 3 CONFIDENTIAL

Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. 10 cm 12 cm 8 cm A). volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh. V = lwh = (10)(12)(8) = 980 cm 3 Next Page: CONFIDENTIAL

B). A cube with edge length 10 cm V= s Volume of a cube = 10 = 1000 cm Substitute 10 for s. 3 Next Page: CONFIDENTIAL

C). A right regular pentagonal Prism with base edge length 5 m and height 7 m. 36° 7 m 5 m Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36° Step:1 tan 36° = 2.5/a The leg of the triangle is half the side length. Or 2.5 m. a = 2.5/tan 36° Solve for a. Next Page: CONFIDENTIAL

Use the value of a to find the base area. 36° 7 m 5 m Step:2 Use the value of a to find the base area. Step:3 Use the base area to find the volume. CONFIDENTIAL

Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon = 0.134 ft ) 3 8 ft 120 ft 60 ft Next Page: CONFIDENTIAL

Find the volume of the aquarium in cubic feet. Step:1 V = lwh = (120)(60)(8) = 57,600 cm 3 8 ft 120 ft 60 ft Next Page: CONFIDENTIAL

8 ft 120 ft 60 ft Step:2 Next Page: CONFIDENTIAL

8 ft 120 ft 60 ft Step:3 The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds CONFIDENTIAL

Cavalieri’s principle also relates to cylinders Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. CONFIDENTIAL

The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h. 2 r h CONFIDENTIAL

Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. 12 cm 8 cm A). Next Page: CONFIDENTIAL

Use the base area to find the radius. A cylinder with a base area of 36 in and a height equal to twice the radius. 2 Step:1 Use the base area to find the radius. 12 cm 8 cm Use the radius to find the height. The height is equal to twice the radius. Step:2 Step:3 Use the radius and height to find the volume. CONFIDENTIAL

Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume. 6 m 12 m CONFIDENTIAL

Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. 5 m 8 m 9 m 6 m CONFIDENTIAL

You did a great job today! CONFIDENTIAL