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Notes 44

Notes 44. Volume of Pyramids and Cones. Suppose you have a square-pyramid-shaped container and a square-prism-shaped container, and the bases and heights are the same size. If you pour sand from the pyramid into the prism, it appears that the prism holds three times as much sand as the pyramid.

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Notes 44

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  1. Notes 44 Volume of Pyramids and Cones

  2. Suppose you have a square-pyramid-shaped container and a square-prism-shaped container, and the bases and heights are the same size. If you pour sand from the pyramid into the prism, it appears that the prism holds three times as much sand as the pyramid.

  3. In fact, the volume of a pyramid is exactly one-third the volume of a prism with the same height and a congruent base. The height of a pyramid is the perpendicular distance from the pyramid’s base to the vertex opposite the base.

  4. Additional Example 1A: Finding the Volume of a Rectangular Pyramid Find the volume of the pyramid to the nearest tenth. Estimate to check whether the answer is reasonable. 1 3 V = Bh Use the formula. 4 ft Find the area of the rectangular base. B = 5 · 7 = 35 1 3 V = · 35 · 4 Substitute for B and h. 7 ft 5 ft Multiply. V ≈ 46.7 ft3 1 3 V = Round the measurements. · 30 · 5 Estimate The answer is reasonable. V = 50 ft3

  5. Additional Example 1B: Finding the Volume of a Rectangular Pyramid Find the volume of the pyramid to the nearest tenth. Estimate to check whether the answer is reasonable. 1 3 V = Bh Use the formula. Find the area of the triangular base. B = ½ · 30 · 25 = 375 1 3 V = · 375 · 26 Substitute for B and h. Multiply. V = 3250 ft3

  6. Check It Out: Example 1A Find the volume of each pyramid to the nearest tenth. Estimate to check whether the answer is reasonable.

  7. Check It Out: Example 1B Find the volume of each pyramid to the nearest tenth. Estimate to check whether the answer is reasonable.

  8. The volume of a cone is one-third the volume of a cylinder with the same height and a congruent base.

  9. 1 3 · ≈ 1. ·52 ·6 V ≈ Additional Example 2A: Finding the Volume of a Cone Find the volume of the cone to the nearest tenth. Use 3.14 for . Estimate to check whether the answer is reasonable. 1 3 Use the formula. The base is a circle so B = · r2 3.14 · 52  78.5. Bh V = 1 3 Substitute for B and h. · 78.5 · 6 V  V  157 yd3 Multiply. 1 3 Estimate  150 yd3 The answer is reasonable.

  10. 7 cm 3 cm 1 3 · ≈ 1. ·32 ·7 V ≈ Additional Example 2B: Finding the Volume of a Cone Find the volume of the cone to the nearest tenth. Use 3.14 for . Estimate to check whether the answer is reasonable. 1 3 Use the formula. The base is a circle so B = · r2 3.14 · 32  28.26. Bh V = 1 3 Substitute for B and h. · 28.26 · 7 V  V  65.9 cm3 Multiply. 1 3 Estimate  63 cm3 The answer is reasonable.

  11. 7 cm 4 cm Check It Out: Example 2 Find the volume of the cone to the nearest tenth. Use 3.14 for . Estimate to check whether the answer is reasonable.

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