Volume of Pyramids and Cones

*Notes 44

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.

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.

Find the volume of the pyramid to the nearest tenth. Estimate to check whether the answer is reasonable.

Additional Example 1A: Finding the Volume of a Rectangular Pyramid

V = 13

Bh Use the formula.

B = 5 · 7 = 35Find the area of the rectangular base.

V = 13

· 35 · 4 Substitute for B and h.

V ≈ 46.7 ft3

Estimate

5 ft 7 ft

4 ft

Multiply.

V = 13

· 30 · 5 Round the measurements.

V = 50 ft3 The answer is reasonable.

Find the volume of the pyramid to the nearest tenth. Estimate to check whether the answer is reasonable.

Additional Example 1B: Finding the Volume of a Rectangular Pyramid

V = 13

Bh Use the formula.

B = ½ · 30 · 25 = 375Find the area of the triangular base.

V = 13 · 375 · 26 Substitute for B and h.

V = 3250 ft3 Multiply.

Find the volume of each pyramid to the nearest tenth. Estimate to check whether the answer is reasonable.

Check It Out: Example 1A

Find the volume of each pyramid to the nearest tenth. Estimate to check whether the answer is reasonable.

Check It Out: Example 1B

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

Find the volume of the cone to the nearest tenth. Use 3.14 for . Estimate to check whether the answer is reasonable.

Additional Example 2A: Finding the Volume of a Cone

V =13 Bh

V 13 · 78.5 · 6

V 157 yd3

Estimate

Use the formula. The base is a circle so B = · r2 3.14 · 52 78.5.

Substitute for B and h.

Multiply.

13

· ≈ 1.13V ≈ · 52 · 6

150 yd3 The answer is reasonable.

Find the volume of the cone to the nearest tenth. Use 3.14 for . Estimate to check whether the answer is reasonable.

Additional Example 2B: Finding the Volume of a Cone

V =13 Bh

V 13 · 28.26 · 7

V 65.9 cm3

Estimate

Use the formula. The base is a circle so B = · r2 3.14 · 32 28.26.

Substitute for B and h.

Multiply.

13

· ≈ 1.13V ≈ · 32 · 7

63 cm3 The answer is reasonable.

7 cm

3 cm

Find the volume of the cone to the nearest tenth. Use 3.14 for . Estimate to check whether the answer is reasonable.

Check It Out: Example 2

7 cm

4 cm