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**Volume of Prisms and Cylinders**

6-6 Volume of Prisms and Cylinders Warm Up Problem of the Day Lesson Presentation Pre-Algebra

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**Learn to find the volume of prisms and cylinders.**

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Vocabulary prism cylinder

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**A prism is a three-dimensional figure named for the shape of its bases**

A prism is a three-dimensional figure named for the shape of its bases. The two bases are congruent polygons. All of the other faces are parallelograms. A cylinder has two circular bases.

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**If all six faces of a rectangular prism are squares, it is a cube. **

Remember! Triangular prism Rectangular prism Cylinder Height Height Height Base Base Base

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**VOLUME OF PRISMS AND CYLINDERS**

Words Numbers Formula Prism: The volume V of a prism is the area of the base B times the height h. Cylinder: The volume of a cylinder is the area of the base B times the height h. B = 2(5) = 10 units2 V = Bh V = 10(3) = 30 units3 B = p(22) V = Bh = 4p units2 = (pr2)h V = (4p)(6) = 24p 75.4 units3

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**Area is measured in square units. Volume is measured in cubic units.**

Helpful Hint

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**Additional Example 1A: Finding the Volume of Prisms and Cylinders**

Find the volume of each figure to the nearest tenth. A. A rectangular prism with base 2 cm by 5 cm and height 3 cm. B = 2 • 5 = 10 cm2 Area of base V = Bh Volume of a prism = 10 • 3 = 30 cm3

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**Additional Example 1B: Finding the Volume of Prisms and Cylinders**

Find the volume of the figure to the nearest tenth. B. B = p(42) = 16p in2 Area of base 4 in. V = Bh Volume of a cylinder 12 in. = 16p • 12 = 192p in3

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**Additional Example 1C: Finding the Volume of Prisms and Cylinders**

Find the volume of the figure to the nearest tenth. B = • 6 • 5 = 15 ft2 1 2 C. Area of base 5 ft V = Bh Volume of a prism = 15 • 7 = 105 ft3 7 ft 6 ft

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Try This: Example 1A Find the volume of the figure to the nearest tenth. A. A rectangular prism with base 5 mm by 9 mm and height 6 mm. B = 5 • 9 = 45 mm2 Area of base V = Bh Volume of prism = 45 • 6 = 270 mm3

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Try This: Example 1B Find the volume of the figure to the nearest tenth. B = p(82) Area of base B. 8 cm = 64p cm2 V = Bh Volume of a cylinder 15 cm = (64p)(15) = 960p 3,014.4 cm3

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Try This: Example 1C Find the volume of the figure to the nearest tenth. C. B = • 12 • 10 1 2 Area of base 10 ft = 60 ft2 V = Bh Volume of a prism 14 ft = 60(14) = 840 ft3 12 ft

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**Additional Example 2A: Exploring the Effects of Changing Dimensions**

A juice box measures 3 in. by 2 in. by 4 in. Explain whether tripling the length, width, or height of the box would triple the amount of juice the box holds. The original box has a volume of 24 in3. You could triple the volume to 72 in3 by tripling any one of the dimensions. So tripling the length, width, or height would triple the amount of juice the box holds.

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**Additional Example 2B: Exploring the Effects of Changing Dimensions**

A juice can has a radius of 2 in. and a height of 5 in. Explain whether tripling the height of the can would have the same effect on the volume as tripling the radius. By tripling the height, you would triple the volume. By tripling the radius, you would increase the volume to nine times the original.

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Try This: Example 2A A box measures 5 in. by 3 in. by 7 in. Explain whether tripling the length, width, or height of the box would triple the volume of the box. The original box has a volume of (5)(3)(7) = 105 cm3. V = (15)(3)(7) = 315 cm3 Tripling the length would triple the volume.

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Try This: Example 2A A box measures 5 in. by 3 in. by 7 in. Explain whether tripling the length, width, or height of the box would triple the volume of the box. The original box has a volume of (5)(3)(7) = 105 cm3. V = (5)(3)(21) = 315 cm3 Tripling the height would triple the volume.

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Try This: Example 2A A box measures 5 in. by 3 in. by 7 in. Explain whether tripling the length, width, or height of the box would triple the volume of the box. The original box has a volume of (5)(3)(7) = 105 cm3. V = (5)(9)(7) = 315 cm3 Tripling the width would triple the volume.

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Try This: Example 2B A cylinder measures 3 cm tall with a radius of 2 cm. Explain whether tripling the radius or height of the cylinder would triple the amount of volume. The original cylinder has a volume of 4 • 3 = 12 cm3. V = 36 • 3 = 108 cm3 By tripling the radius, you would increase the volume nine times.

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Try This: Example 2B A cylinder measures 3 cm tall with a radius of 2 cm. Explain whether tripling the radius or height of the cylinder would triple the amount of volume. The original cylinder has a volume of 4 • 3 = 12 cm3. V = 4 • 9 = 36 cm3 Tripling the height would triple the volume.

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**Additional Example 3: Construction Application**

A section of an airport runway is a rectangular prism measuring 2 feet thick, 100 feet wide, and 1.5 miles long. What is the volume of material that was needed to build the runway? length = 1.5 mi = 1.5(5280) ft = 7920 ft The volume of material needed to build the runway was 1,584,000 ft3. width = 100 ft height = 2 ft V = 7920 • 100 • 2 ft3 = 1,584,000 ft3

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Try This: Example 3 A cement truck has a capacity of 9 yards3 of concrete mix. How many truck loads of concrete to the nearest tenth would it take to pour a concrete slab 1 ft thick by 200 ft long by 100 ft wide? B = 200(100) = 20,000 ft2 V = 20,000(1) = 20,000 ft3 20,000 27 yd3 27 ft3 = 1 yd3 740.74 9 = 82.3 Truck loads

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**Additional Example 4: Finding the Volume of Composite Figures**

Find the volume of the the barn. Volume of barn Volume of rectangular prism Volume of triangular prism + = V = (40)(50)(15) (40)(10)(50) 1 2 = 30, ,000 = 40,000 ft3 The volume is 40,000 ft3.

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**Find the volume of the figure.**

Try This: Example 4 Find the volume of the figure. Volume of barn Volume of rectangular prism Volume of triangular prism + = = (8)(3)(4) (5)(8)(3) 1 2 5 ft = 4 ft V = 156 ft3 8 ft 3 ft

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Challenge: A 6 cm section of plastic water pipe has inner diameter 12 cm and outer diameter 15 cm. Find the volume of the plastic pipe, not the hollow interior, to the nearest tenth.

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Lesson Quiz Find the volume of each figure to the nearest tenth. Use 3.14 for p. 10 in. 1. 2. 3. 2 in. 12 in. 12 in. 10.7 in. 15 in. 3 in. 8.5 in. 942 in3 306 in3 160.5 in3 4. Explain whether doubling the radius of the cylinder above will double the volume. No; the volume would be quadrupled because you have to use the square of the radius to find the volume.

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