# Volumes of Pyramids and Cones

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Volumes of Pyramids and Cones
10.8 LESSON Volumes of Pyramids and Cones Consider a prism and a pyramid that have the same base area and the same height. If you completely fill the pyramid with sand and pour the sand into the prism, you’ll find that the sand fills one third of the prism. You can conclude that the volume of the pyramid is one third of the volume of the prism. The same relationship holds for a cylinder and a cone with the same base area and the same height.

Volumes of Pyramids and Cones
10.8 LESSON Volumes of Pyramids and Cones Volume of a Pyramid or a Cone Words The volume V of a pyramid or a cone is one third of the product of the base area B and the height h. V = Bh 1 3 Algebra

Volumes of Pyramids and Cones
10.8 LESSON Volumes of Pyramids and Cones EXAMPLE 1 Finding the Volume of a Pyramid The base of a pyramid is a square. The side length of the square is 24 feet. The height of the pyramid is 9 feet. Find the volume of the pyramid. V = Bh 1 3 Write formula for volume of a pyramid. = (24 2)(9) 1 3 Substitute 24 2 for B and 9 for h. = 1728 Simplify. ANSWER The volume of the pyramid is 1728 cubic feet.

Volumes of Pyramids and Cones
10.8 LESSON Volumes of Pyramids and Cones EXAMPLE 2 Finding the Volume of a Cone Find the volume of the cone shown. Round to the nearest cubic millimeter. The radius is one half of the diameter, so r = 6.75. V = πr 2h 1 3 Write formula for volume of a cone. = π(6.75) 2(10) 1 3 Substitute 6.75 for r and 10 for h. ≈ 477.1 Evaluate. Use a calculator. ANSWER The volume of the cone is about 477 cubic millimeters.

Volumes of Pyramids and Cones
10.8 LESSON Volumes of Pyramids and Cones EXAMPLE 3 Finding the Volume of a Solid Silos The grain silo shown is composed of a cylinder and a cone. Find the volume of the silo to the nearest cubic foot. 1 Find the volume of the cylindrical section. The radius is one half of the diameter, so r = 9. V = πr 2h Write formula for volume of a cylinder. = π(9) 2(29) = 2349π Substitute values. Then simplify. 2 Find the volume of the conical section. V = πr 2h 1 3 Write formula for volume of a cone. = π(9) 2(7) 1 3 = 189π Substitute 9 for r and 7 for h. Then simplify.

Volumes of Pyramids and Cones
10.8 LESSON Volumes of Pyramids and Cones EXAMPLE 3 Finding the Volume of a Solid Silos The grain silo shown is composed of a cylinder and a cone. Find the volume of the silo to the nearest cubic foot. 3 Find the sum of the volumes. 2349π + 189π = 2538π ≈ 7973.4 ANSWER The volume of the silo is about 7973 cubic feet.

Volumes of Pyramids and Cones
10.8 LESSON Volumes of Pyramids and Cones Surface Areas and Volumes of Solids SUMMARY Prism Surface Area S = 2B + Ph Cylinder Surface Area S = 2πr 2 + 2πrh Volume V = Bh Volume V = πr 2h Pyramid Surface Area S = B + Pl 1 2 Cone Surface Area S = πr 2 + πrl Volume V = Bh 1 3 Volume V = πr 2h 1 3

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