Volume of Pyramids and Cones. Tape together as shown. Tape together as shown.
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1 7-6 Volume of Pyramids and Cones MAIN IDEA Find the volumes of pyramids and cones. New Vocabulary cone Math Online glencoe.com Extra Examples Personal Tutor Self-Check Quiz In this Mini Lab, you will investigate the relationship between the volume of a pyramid and the volume of a prism with the same base area and height. Draw and cut out 5 squares. 2 in. Draw and cut out 4 isosceles triangles. Tape together as shown. Tape together as shown. Fold and tape to form a cube with an open top. Fold and tape to form an open square pyramid. 1 in. 2 in in. 1. Compare the base areas and the heights of the two solids. 2. Fill the pyramid with rice, sliding a ruler across the top to level the amount. Pour the rice into the cube. Repeat until the prism is filled. How many times did you fill the pyramid in order to fill the cube?. What fraction of the cube s volume does one pyramid fill? The volume of a pyramid is one-third the volume of a prism with the same base area and height. Volume of a Pyramid Key Concept Words The volume V of a pyramid is one-third the area of the base B times the height h. Symbols V = Bh Model The height of a pyramid or cone is the distance from the vertex, perpendicular to the base. 80 Chapter 7 Measurement: Area and Volume 80_84_C07_L06_ indd 80 9/22/07 12:44:44 PM
2 Find the Volume of a Pyramid Estimation You can estimate the volume of the pyramid in Example 1 to be about ( 2 8 6) (11) or 88 m. Since m is close to 88 m, the answer is reasonable. 1 Find the volume of the pyramid. Round to the nearest tenth. V = _ 1 Bh Volume of a pyramid 11 m V = _ , h = 11 ( 1 _ V = ) 11 B = 2 Simplify. The volume is about 95.0 cubic meters. 8.1 m 6.4 m a. Find the volume of a pyramid that has a height of 5 yards and a square base with sides 2 yards long. 2 ARCHITECTURE The Louvre Pyramid in Paris has a square base with sides 112 feet long. If the volume is 296,875 cubic feet, find the height of the pyramid. V = _ 1 Bh Volume of a pyramid 296,875 = _ 1 12,544 h Replace V with 296,875 and B with or 12, ,875 = _ 12,544 h Multiply. 12,544 _ 12,544 h Multiply each side by 12, ,875 = 71 h Simplify. The height of the pyramid is about 71 feet. _ 12,544. b. CRAFTS Nicco made a pyramid-shaped candle. The volume of the candle is 864 cubic centimeters and its base has an area of 144 square centimeters. How high is the candle? A cone is a three-dimensional figure with one circular base. A curved surface connects the base and the vertex. The volumes of a cone and a cylinder are related in the same way as those of a pyramid and prism. Volume of a Cone Key Concept Words Symbols V = The volume V of a cone with radius r is one-third the area of the base B times the height h. Bh or V = π r 2 h Model Lesson 7-6 Volume of Pyramids and Cones 81 80_84_C07_L06_ indd 81 9/22/07 12:44:47 PM
3 Find the Volume of a Cone Find the volume of the cone. mm V = _ 1 π r 2 h Volume of a cone V = _ 1 π 2 14 Replace r with and h with 14. V 11.9 Simplify. Use a calculator. The volume is about 11.9 cubic millimeters. 14 mm c. d. 17 ft 4 ft 10 cm Example 1 (p. 81) Find the volume of each pyramid. Round to the nearest tenth cm in. 14 cm in. 10 in.. Find the volume of a pyramid that has a height of 125 centimeters and a square base with sides 95 centimeters long. 4. Find the volume of a pyramid that has a height of 17 feet and a square base with sides 22 feet long. Example 2 (p. 81) Example (p. 82) 5. ARCHAEOLOGY El Castillo, the pyramid at Chichen Itza in Mexico is 0 meters tall with a volume of about 0,580 cubic meters. What is the length of each side of the square base? 6. 7 m 5 m in. 11 in yd cm 4 yd 15 cm 82 Chapter 7 Measurement: Area and Volume 80_84_C07_L06_ indd 82 9/22/07 12:44:49 PM
4 For Exercises See Examples HELP HOMEWORK HELP Find the volume of each pyramid. Round to the nearest tenth in. 4 in yd 1 yd 15 yd triangular pyramid: triangle base, 10 cm; triangle height, 7 cm; pyramid height, 15 cm mi ft 9 ft 15 mm 21 mm 2.5 mi 17. cone: diameter, 12 m; height, 5 m 18. SCIENCE A model of a volcano constructed for a science project is cone-shaped with a diameter of 8 inches. If the volume of the model is about 201 cubic inches, how tall is the model? Find the volume of each solid. Round to the nearest tenth if necessary yd m mm 6 mm m 8 yd 15 yd 6 yd 4 m 2 m 5 mm 22. HATS A clown wants to fill his party hat with confetti. Use the drawing at the right to determine how much confetti his hat will hold. 6 in. 8 in. PRACTICE EXTRA PRACTICE See pages 687, IRRIGATION A water tank like the one at the right is used to water flowers at a park. Water can be pumped from the tank at a rate of 25 liters per minute. How long will it take to use all of the water in a full tank? Round to the nearest minute. (Hint: 1 liter = 1,000 cm ) 0.5 m 0.25 m 1.5 m Lesson 7-6 Volume of Pyramids and Cones 8 80_84_C07_L06_ indd 8 9/22/07 12:44:51 PM
5 H.O.T. Problems 24. CHALLENGE How could you change the height of a cone so that its volume would remain the same when its radius was tripled? 25. OPEN ENDED Draw and label a triangular pyramid with a volume of 6 cubic centimeters. 26. NUMBER SENSE Which would have a greater effect on the volume of a cone, doubling its radius or doubling its height? Explain your reasoning. WRITING IN 27. MATH Write about a real-world situation that can be solved by finding the volume of a cone. 28. A rectangular pyramid has a base 18 inches by 0 inches and a height of 6 inches. Which is closest to the volume of the pyramid in cubic feet? A 2.5 ft B ft C 4 ft D 5.5 ft 29. Find the volume of the cylinder. Round to the nearest tenth if necessary. 8ft 6ft F 48 ft H 288 ft G 150. ft J ft 0. DISPENSER Find the volume of the soap dispenser at the right. (Lesson 7-5) 1. Name the number and shapes of the faces of a trapezoidal prism. Then name the number of edges and vertices. (Lesson 7-4) 7 in. in. 2. GEOMETRY Graph triangle ABC with vertices A(1, 2), B(4, -1), and C(2, -4). Then graph its image after a reflection over the y-axis, and write the coordinates of the image s vertices. (Lesson 6-6). SHOPPING Etu saved $90 when he purchased a DVD recorder on sale. If the sale price was 7.5% off the regular price, what was the regular price of the DVD recorder? (Lesson 5-7) PREREQUISITE SKILL Find the circumference of each circle. Round to the nearest tenth. (Lesson 7-1) 4. diameter, 9 in. 5. diameter, 5.5 ft 6. radius, 2 m 7. radius,. 84 Chapter 7 Measurement: Area and Volume 80_84_C07_L06_ indd 84 9/22/07 12:44:5 PM
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