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13. 2 Volumes of Pyramids and Cones

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Objectives: Find the volumes of pyramids. Find the volumes of pyramids. Find the volumes of cones. Find the volumes of cones.

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Pyramids In geometry, a pyramid is a solid shape with triangular side-faces meeting at a common vertex and with a polygon as its base. Pyramids are generally classified by their bases. A pyramid with a square base is called a Square Pyramid, with a triangle base it is called a Tetrahedron, pentagon base and it’s a Pentagonal pyramid. And so on… If the altitude (height) is lined up in the center of the pyramid, it is called a Right Pyramid. Otherwise it is known as an Oblique Pyramid. Height Right PyramidOblique Pyramid

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Volume of a Pyramid If a pyramid has a volume of V cubic units, a height of h units, and a base with an area of B square units, then… 1 3 1 3 V = Bh B equaling the area of the base, h equaling the height of the pyramid, and V being the total volume Why is it ? 1 3 Remember that the volume of a prism is V=Bh. If the area of the base of the pyramid was equal to the bases of a prism and the heights of the pyramid and prism were also equal, then the volume of the prism would be three times that of a pyramid. The same goes for cones and cylinders.

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Example 1: Travis is making a plaster model of the Food Guide Pyramid for a class presentation. The model is a square pyramid with a base edge of 12 inches and a height of 15 inches. Find the volume of a plaster needed to make the model. 12 15

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Example 1 Continued: = s h = s h = (12 )(15) = (12 )(15) = 720 = 720 Volume of a Pyramid B = s s = 12, h = 15 Simplify 1 3 1 3 V = Bh 1 3 1 3 22 1 3 1 3 2 Travis needs 720 cubic inches of plaster to make his model.

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In geometry, a cone is a pyramid with a circular base. Right Cones If the vertex is directly above the center of the circle, it is known as a right circular cone. Oblique Cones When the vertex of a Cone is not aligned directly above the center of its base, it is called an oblique cone. Cones

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Volume of a Cone If a circular cone – right or oblique – has a volume of V cubic units, a height of h units, and the base has a radius of r units, then… V = Bh 1 3 Since the base of a cone is a circle you could also use: V = π r h 1313 2 B equaling the area of the base, h equaling the height of the cone, and V being the total volume.

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Example 2: Find the Volume of the Right Cone 8 in

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Example 2 Continued: = (8 )(8) = π (8 )(8) = ((64)(8) = ( π / 3)(64)(8) = ((512) = ( π / 3)(512) = 536.165 = 536.165 Volume of a Cone radius = 8, height = 8 Multiply V = π r h 1 3 2 1313 2 3 The volume of the cone is approximately 536.2 inches

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Example 3: Find the volume of the cone. 10 in 48 o 10 in 48 o

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Example 3 Continued: First we must find the radius. tan A = opposite adjacent tan 48 = 10 r r = 10__ tan 48 r = 9.0 Now the volume. V = π r h 2 1313 1313 = π (9 )(10) o 2 o )(81)(10) = ( π / 3 )(81)(10) )(810) = ( π / 3 )(810) = 848.992 The volume of the cone is approximately 849.0 units 3

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¡Assignment! Pg. 699 8 – 18, 20, 25 – 27, 31 ☺ #19 is Extra Credit

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