OBJECTIVEAFTER STUDYING THIS SECTION, YOU WILL BE ABLE TO FIND THE VOLUMES

OF PYRAMIDS AND CONES. YOU WILL BE ABLE TO SOLVE PROBLEMS INVOLVING CROSS SECTIONS OF PYRAMIDS AND

CONES

12.5 Volume of Pyramids and Cones

Volume of Pyramids

The volume of a pyramid is related to the volume of a prism having the same base and height.

The volume of a pyramid is one third of the volume of a prism with the same base and height.

Theorem

The volume of a pyramid is equal to one third of the product of the height and the area of the base

where B is the area of the base and h is the height.

1

3pyrV Bh

Cone

The cone is a close relative of the pyramid. Although the base is circular rather than polygonal, the formula for its volume is similar to the formula for the pyramid’s volume.

Slant height (italicized l)height

radius

Theorem

The volume of a cone is equal to one third of the produce of the height and the area of the base.

where B is the area of the base, h is the height, and r is the radius of the base.

21 1

3 3coneV Bh r h

Cross Section of a Pyramid or a Cone

Unlike a cross section of a prism or a cylinder, a cross section of a pyramid or cone is not congruent to the figure’s base. The cross section parallel to the base is similar to the base.

Theorem

In a pyramid or a cone, the ratio of the area of a cross section to the area of the base equals the square of the ration of the figures’ respective distances from the vertex.

Where A is the area of the cross section, B is the area of the base, and k is the distance from the vertex to the cross section, and h is the height of the pyramid or cone.

2A k

B h

A

B

kh

Example 1

If the height of a pyramid is 21 and the pyramid’s base is an equilateral triangle with sides measuring 8, what is the pyramid’s volume?

Example 2

Find the volume of a cone with a base radius of 6 and a slant height of 10.

6

10

h

Example 3

A pyramid has a base area of 24 sq. cm and a height of 12 cm. A cross section is cut 3 cm from the base.

A. Find the volume of the upper pyramid (the solid above the cross section)

B. Find the volume of the frustum (the solid below the cross section).

12 cm

Summary

Explain in your own words how to find the volume of a pyramid and cone.

Homework: worksheet