21.2 Volume of Pyramids
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1 Name lass ate 21.2 Volume of Pyramids Essential Question: How do you find the volume of a pyramid? Explore eveloping a Volume Formula You can think of irregular pyramids as parts of a rectangular prism. This cube can be divided into three square pyramids. Resource Locker The volume of a pyramid is related to the volume of a prism with the same base and height. This triangular pyramid can be thought of as part of a triangular prism. E F To find the volume of the first pyramid, -, first let the area of the base of be, and let the height of the pyramid,, be h. The edges _ E and _ F are congruent to _ and parallel to _. The bases of the prism, EF and, are congruent. What is the volume of the triangular prism in terms of and h? Module Lesson 2
2 You will now compare the volume of pyramid - and the volume of the triangular prism. _ raw E. This is the diagonal of a rectangle, so. Explain why pyramids -E and -FE have the same volume. E E F F Explain why pyramids -EF and - have the same volume. E E F F You have now shown that the three pyramids that form the triangular prism all have the same volume. ompare the volume of pyramid - and the volume of the triangular prism. Write the volume of pyramid - in terms of and h. Module Lesson 2
3 Reflect 1. Explain how you know that the three pyramids that form that triangular prism all have the same volume. Explain 1 Finding the Volume of a Pyramid Since the volume of any triangular or rectangular pyramid can be shown to compose one third the volume of a rectangular prism, the formula for the volume of a triangular prism can be generalized. Volume of a Pyramid The volume V of a pyramid with base area and height h is given by V= 1 h. Example 1 Solve a volume problem. shton built a model square-pyramid with the dimensions shown. What is the volume of the pyramid? The pyramid is composed of wooden blocks that are in the shape of cubes. block has the dimensions 4 cm by 4 by 4 cm. How many wooden blocks did shton use to build the pyramid? 16 cm 24 cm Find the volume of the pyramid. 24 cm The area of the base is the area of the square with sides of length 24 cm. So, = 576 cm 2. The volume V of the pyramid is _ 1 h = 1_ So V = 072 c m. Find the volume of an average block. The volume of a cube is given by the formula V = s. So the volume W of a wooden block is 64 c m. Find the approximate number of stone blocks in the pyramid, divide V by W. So the number of blocks that shton used is 144. Module Lesson 2
4 The Great Pyramid in Giza, Egypt, is approximately a square pyramid with the dimensions shown. The pyramid is composed of stone blocks that are rectangular prisms. n average block has dimensions 1. m by 1. m 146 m by 0.7 m. pproximately how many stone blocks were used to build the 20 m pyramid? Round to the 20 m nearest hundred thousand. Find the volume of the pyramid. The area of the base is the area of the square with sides of length 20 m. So,. The volume V of the pyramid is _ 1 h = 1_. So V. Find the volume of an average block. The volume of a rectangular prism is given by the formula. So the volume W of an average block is. Find the approximate number of stone blocks in the pyramid, divide by. So the approximate number of blocks is. Reflect 2. What aspects of the model in Part may lead to inaccuracies in your estimate?. Suppose you are told that the average height of a stone block 0.69 m rather than 0.7 m. Would the increase or decrease your estimate of the total number of blocks in the pyramid? Explain. Your Turn 4. piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 19.7 grams. What is the density of silver? Round to the nearest tenth. ( Hint: density = mass volume. ) 2.5 cm 1.5 cm 1.5 cm Image redits: Mark Goddard/iStockPhoto.com Module Lesson 2
5 Explain 2 Finding the Volume of a omposite Figure You can add or subtract to find the volume of composite figures. Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism. Round to the nearest tenth. 15 ft 25 ft 12 ft Find the volume of the prism. V = lwh = (25) (12) ( 15 ) = 4500 ft Find the volume of pyramid. rea of base: = (25) ( 12 ) = 00 ft 2 Volume of pyramid: V = 1_ (00)(15) = 1500 ft Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure = 000 So the volume of the composite figure is 000 ft. 15 cm Find the volume of the prism. 0 cm V = lwh = (0) ( ) ( ) = ( ) cm Find the volume of the pyramid. rea of base: = cm 2 12 cm Volume of pyramid: V = 1_ ( )( ) = ( ) cm Subtract volume of pyramid from volume of prism to find volume of composite figure. - = So the volume of the composite figure is cm. Module Lesson 2
6 Your Turn Find the volume of the composite figure. Round to the nearest tenth. 5. The composite figure is formed from two pyramids. The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches. 6. The composite figure is formed by a rectangular prism with two square pyramids on top of it. ft 2 ft 10 ft 5 ft Elaborate 7. Explain how the volume of a pyramid is related to the volume of a prism with the same base and height. 8. If the length and width of a rectangular pyramid are doubled and the height stays the same, how does the volume of the pyramid change? Explain. 9. Essential Question heck-in How do you calculate the volume of a pyramid? Module Lesson 2
7 Evaluate: Homework and Practice 1. ompare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid. Online Homework Hints and Help Extra Practice 2. Which of the following equations could describe a square pyramid? Select all that apply.. Vh =. V = 1_ lw. w = V_ lh. V_ = h_ E. V = _ w 2 h F. 1_ = Vh. Justify Reasoning s shown in the figure, polyhedron EFGH is a cube and P is any point on face EFGH. ompare the volume of the pyramid P and the volume of the cube. emonstrate how you came to your answer. E H P F G Find the volume of the pyramid. Round your answer to the nearest tenth mm mm 12.5 mm 17 in. 6 in. 4 in. Module Lesson 2
8 6. Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft. 7. The area of the base of a hexagonal pyramid 24 is tan 0 cm 2. Find its volume. 4 cm 4 cm Find the volume of the composite figure. Round to the nearest tenth cm 7.5cm 12 cm 5 cm 25 cm 12.5 cm 12 cm 12 cm 10. Given a square pyramid with a height of 21 ft and a volume of 969 cubic feet, find the length of one side of the square base. Round to the nearest tenth. 11. onsider a pyramid with height 10 feet and a square base with side length of 7 feet. How does the volume of the pyramid change if the base stays the same and the height is doubled? Module Lesson 2
9 12. lgebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters. 1. Find the height of a rectangular pyramid with length meters, width 8 meters, and volume 112 cubic meters. X 10 cm 10 cm 14. storage container for grain is in the shape of a square pyramid with the dimensions shown. 1 m a. What is the volume of the container in cubic centimeters? b. Grain leaks from the container at a rate of 4 cubic centimeters per second. ssuming the container starts completely full, about how many hours does it take until the container is empty? 1.5 m 15. piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of grams. What is the density of copper? Round to the nearest hundredth. ( Hint: density = _ mass volume. ) 2.5 cm 1.5 cm 1.5 cm Module Lesson 2
10 16. Represent Real World Problems n art gallery is a 6 story square pyramid with base area 1 2 acre (1 acre = 4840 yd 2, 1 story 10 ft). Estimate the volume in cubic yards and cubic feet. 17. nalyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6? Explain how you found your answer. 7 ft 7 ft 4 ft 18. Geology crystal is cut into a shape formed by two square pyramids joined at the base. Each pyramid has a base edge length of 5.7 mm and a height of mm. What is the volume of the crystal to the nearest cubic millimeter? 19. roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards. What is the volume of the attic in cubic feet? In cubic yards? mm 5 yd 5.7 mm 45 ft Module Lesson 2
11 H.O.T. Focus on Higher Order Thinking 20. Explain the Error escribe and correct the error in finding the volume of the pyramid. V = 1(49)(10) 2 = 245 ft 10 ft 7 ft 21. ommunicate Mathematical Ideas pyramid has a square base and a height of 5 ft. The volume of the pyramid is 60 ft. Explain how to find the length of a side of the pyramid s base. 22. ritical Thinking triangular pyramid has a length of 2, a width of x, and a height of x. Its volume is 512 cm. What is the area of the base? Module Lesson 2
12 Lesson Performance Task Genna is making a puzzle using a wooden cube. She s going to cut the cube into three pieces. The figure below shows the lines along which she plans to cut away the first piece. The result will be a piece with four triangular sides and a square side (shaded). 1. Each cut Genna makes will begin at the upper left corner of the cube. Write a rule describing where she drew the lines for the first piece. 2. The figure below shows two of the lines along which Genna will cut the second piece. raw a cube and on it, draw the two lines Genna drew. Then, using the same rule you used above, draw the third line and shade the square base of the second piece.. When Genna cut away the second piece of the puzzle, the third piece remained. raw a new cube and then draw the lines that mark the edges of the third piece. Shade the square bottom of the third piece. 4. ompare the volumes of the three pieces. Explain your reasoning. 5. Explain how the model confirms the formula for the volume of a pyramid. Module Lesson 2
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