8-9 Volume of Pyramids and Cones

Lesson PresentationLesson Presentation

8-9 Volume of Pyramids and Cones

MA.7.G.2.1 Justify and apply formulas for… volume of pyramids…and cones.Also MA.7.G.2.2

Sunshine State Standards

8-9 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.

8-9 Volume of Pyramids and Cones

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.

8-9 Volume of Pyramids and Cones

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

Additional Example 1: 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.

8-9 Volume of Pyramids and Cones

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

Check It Out: Example 1

V = 13

Bh Use the formula.

B = 8 · 7 = 56Find the area of the rectangular base.

V = 13

· 56 · 4 Substitute for B and h.

V ≈ 74.7 ft3

Estimate

Multiply.

V = 13

· 60 · 4 Round the measurements.

V = 80 ft3 The answer is reasonable.

8 ft 7 ft

4 ft

8-9 Volume of Pyramids and Cones

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

8-9 Volume of Pyramids and Cones

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.

8-9 Volume of Pyramids and Cones

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

8-9 Volume of Pyramids and Cones

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 2A

V =13 Bh

V 13 · 12.56 · 3

V = 12.56 yd3

Estimate

Use the formula. The base is a circle so B = · r2 3.14 · 22 12.56.

Substitute for B and h.

Multiply.

13

· ≈ 1.13V ≈ · 22 · 3

12 yd3 The answer is reasonable.

3 yd

2 yd

8-9 Volume of Pyramids and Cones

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 2B

V =13 Bh

V 13 · 12.56 · 7

V 29.3 cm3

Estimate

Substitute for B and h.

Multiply.

13

· ≈ 1.13V ≈ · 22 · 7

28 cm3 The answer is reasonable.

7 cm

4 cm

Use the formula. The base is a circle so B =

· r2 3.14 · 2 12.56.4 2