Volume of Rectangular Prisms and Pyramids. Use the formula. Substitute for l and w. Use the formula. Substitute for B and h.

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1 ? LESSON 10.1 ESSENTIAL QUESTION Volume of Rectangular Prisms and Pyramids How do you find the volume of a rectangular prism and a rectangular pyramid? Finding the Volume of a Rectangular Prism Remember that the volume of a rectangular prism is given by the formula V = l w h, or V = lwh. The base of a rectangular prism is a rectangle with length l and width w, so the area of the base B is equal to lw. The volume formula can also be written as. In fact, the volume of any prism is the product of the base B and the height h. Remember that volume is given in cubic units. Equations, expressions, and relationships 7.8.A Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas. Also 7.9.A Math On the Spot Volume of a Prism The volume V of a prism is the area of its base B times its height h. Image Credits: Photodisc/ Getty Images EXAMPLE 1 Find the volume of the rectangular prism. STEP 1 STEP 2 Find the area of the base. B = lw B = B = 180 m m 2 Find the volume. Use the formula. Substitute for l and w. Use the formula. 12 mm 15 mm 7 mm 7.9.A Math Talk Mathematical Processes Does it matter which dimension of a rectangle you consider its height? V = Substitute for B and h. V = 1,260 m m 3 The volume of the rectangular prism is 1,260 cubic millimeters. Lesson

2 Reflect 1. What If? If you know the volume V and the height h of a prism, how would you find the area of the base B? YOUR TURN Personal Math Trainer Online Assessment and Intervention 2. Use the formula to find the volume of a gift box that is 3.5 inches high, 7 inches long, and 6 inches wide. EXPLORE ACTIVITY A Nets A net is a two-dimensional pattern of shapes that can be folded into a three-dimensional figure. The shapes in the net become the faces of the three-dimensional figure. STEP 1 Copy Net A and Net B on graph paper, and cut them out along the blue lines. Net A Net B One of these nets can be folded along the black lines to make a cube. Which net will not make a cube? STEP 2 See if you can find another net that can be folded into a cube. Draw a net that you think will make a cube on your graph paper, and then cut it out. Can you fold it into a cube? Sketch your net below. STEP 3 Compare your results with several of your classmates. How many different nets for a cube did you and your classmates find? 316 Unit 5

3 Reflect 3. What shapes will appear in a net for a rectangular prism that is not a cube? How many of these shapes will there be? How do you know that each net cannot be folded into a cube without actually cutting and folding it? Make a Conjecture If you draw a net for a cylinder, such as a soup can, how many two-dimensional geometric shapes would this net have? Name the shapes in the net for a cylinder. Exploring the Volume of a Rectangular Pyramid A pyramid is a three-dimensional shape whose base is a polygon and whose other faces are all triangles. Like a prism, a pyramid is named by the shape of its base. Rectangular Pyramid Triangular Pyramid Pentagonal Pyramid The faces of a pyramid that are not the base have a common vertex, called the vertex of the pyramid. The perpendicular distance from the vertex to the base is the height of the pyramid. Height Base Lesson

4 EXPLORE ACTIVITY A In this activity, you will compare the volumes of a pyramid and a prism with congruent bases and equal heights. Remember that congruent figures have the same shape and size. STEP 1 Make three-dimensional models. Make larger versions of the nets shown. Make sure the bases and heights in each net are the same size. Fold each net, and tape it together to form a prism or a pyramid. STEP 2 Fill the pyramid with beans. Make sure that the beans are level with the opening of the pyramid. Then pour the beans into the prism. Repeat until the prism is full. How many times did you fill the prism from the pyramid? STEP 3 Write a fraction that compares the volume of the pyramid to the volume of the prism. volume of pyramid = volume of prism Math Talk Mathematical Processes Describe ways in which a prism and a pyramid are different. Reflect 8. Communicate Mathematical Ideas The prism and the pyramid in this activity have congruent bases and equal heights. Are they congruent three-dimensional shapes? Explain. YOUR TURN Personal Math Trainer Online Assessment and Intervention 318 Unit The volume of a rectangular prism is 4_12 in. What is the volume of a rectangular pyramid with a congruent base and the same height? Explain your reasoning. 7. Draw Conclusions A rectangular pyramid has a base area of B and a height of h. What is a formula for the volume of the pyramid? Justify your reasoning.

5 Solving Volume Problems You can use the formulas for the volume of a rectangular prism and the volume of a rectangular pyramid to solve problems. Volume of a Rectangular Pyramid Math On the Spot The volume V of a pyramid is one-third the area of its base B times its height h. V = 1_ 3 Bh EXAMPLE A A Kyle needs to build a crate in the shape of a rectangular prism. The crate must have a volume of 38 1_ 2 cubic feet, and a base area of 15 2_ 5 square feet. Find the height of the crate. 38 1_ 2 = 15 2_ 5 h Use the formula. Substitute for V and B. B 77 2 = 77 5 h = 77 5 h _ 2 = h The height of the crate must be 5 _ 2, or 2 1_ 2, feet. A glass paperweight in the shape of a rectangular pyramid has a base that is 4 inches by 3 inches and a height of 5 inches. Find the volume of the paperweight. V = 1_ 3 Bh V = 1_ V = 20 The paperweight has a volume of 20 cubic inches. YOUR TURN Change the mixed numbers to fractions. To divide both sides by 77, multiply both 5 sides by the reciprocal. Use the formula. Think: B = lw = 4 3 = A rectangular prism has a volume of 160 cubic centimeters and a height of 4 centimeters. What is the area of its base? 11. A square pyramid has a base edge of 5.5 yards and a height of 3.25 yards. Find the volume of the pyramid to the nearest tenth. Personal Math Trainer Online Assessment and Intervention Lesson

6 Guided Practice 1. Find the volume of the rectangular prism. (Example 1) ( V = V = ft 3 ) ( ) 6 ft 9 ft ft Identify the three-dimensional shape that can be formed from each net. (Explore Activity 1 and Explore Activity 2) The volume of a rectangular prism is m 3. The prism has a base that is 5.2 m by 3.1 m. Find the height of the prism. (Example 2) = ( ) h = h? ESSENTIAL QUESTION CHECK-IN 6. Explain how to use models to show the relationship between the volume of a rectangular prism and a rectangular pyramid with congruent bases and heights. = h The height of the prism is. 320 Unit 5

7 Name Class Date 10.1 Independent Practice 7.8.A, 7.9.A Personal Math Trainer Online Assessment and Intervention 8. Explain the Error A student found the volume of a rectangular pyramid with a base area of 92 square meters and a height of 54 meters to be 4,968 cubic meters. Explain and correct the error. 13. A storage chest has the shape of a rectangular prism with the dimensions shown. The volume of the storage chest is 18,432 cubic inches. What is its height? 9. A block of marble is in the shape of a rectangular prism. The block is 3 feet long, 2 feet wide, and 18 inches high. What is the 48 in. 16 in. volume of the block? 10. Multistep Curtis builds a doghouse with base shaped like a cube and a roof shaped like a pyramid. The cube has an edge length of 3 1_ feet. The height of the pyramid 2 is 5 feet. Find the volume of the doghouse rounded to the nearest tenth. 14. Draw Conclusions A shipping company ships certain boxes at a special rate. The boxes must not have a volume greater than 2,500 cm 3. Can the box shown be shipped at the special rate? Explain. 20 cm 12 cm 10 cm 11. Miguel has an aquarium in the shape of a rectangular prism. The base is inches long and 12.5 inches wide. The aquarium is inches high. What is the volume of the aquarium to the nearest cubic inch? 12. After a snowfall, Sheree built a snow pyramid. The pyramid had a square base with side lengths of 32 inches and a height of 28 inches. What was the volume of the pyramid to the nearest cubic inch? 15. Communicate Mathematical Ideas Is the figure shown a prism or a pyramid? Justify your answer. 4 in. 11 in. 8 in. 9 in. Lesson

8 16. A cereal box can hold 144 cubic inches of cereal. Suppose the box is 8 inches long and 1.5 inches wide. How tall is the box? 17. A shed in the shape of a rectangular prism has a volume of 1,080 cubic feet. The height of the shed is 8 feet, and the width of its base is 9 feet. What is the length of the shed? FOCUS ON HIGHER ORDER THINKING Work Area 18. Draw Conclusions Sue has a plastic paperweight shaped like a rectangular pyramid. The volume is 120 cubic inches, the height is 6 inches, and the length is 10 inches. She has a gift box that is a rectangular prism with a base that is 6 inches by 10 inches. How tall must the box be for it to hold the pyramid? 19. Represent Real-World Problems A public swimming pool is in the shape of a rectangular prism. The pool is 20 meters long and 16 meters wide. The pool is filled to a depth of 1.75 meters. a. Find the volume of water in the pool. b. A cubic meter of water has a mass of 1,000 kilograms. Find the mass of the water in the pool. 20. Analyze Relationships There are two glass pyramids at the Louvre Museum in Paris, France. The outdoor pyramid has a square base with side lengths of 35.4 meters and a height of 21.6 meters. The indoor pyramid has a square base with side lengths of 15.5 meters and a height of 7 meters. How many times as great is the volume of the outdoor pyramid than that of the indoor pyramid? 21. Persevere in Problem Solving A small solid pyramid was installed on top of the Washington Monument in The square base of the pyramid is 13.9 centimeters on a side, and the height of the pyramid is 22.6 centimeters. The pyramid has a mass of 2.85 kilograms. a. Find the volume of the pyramid. Round to the nearest hundredth. b. Find the mass of the pyramid in grams. c. Science The density of a substance is the ratio of its mass to its volume. Find the density of the pyramid in grams per cubic centimeter. Round to the nearest hundredth. 322 Unit 5

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