Section 14.5: Volumes of Pyramids

Property

A pyramid has exactly one-third the volume of a prism with the same base and height: $V_{pyramid} = \frac{1}{3} \times V_{prism}$

Examples

Property

The Pythagorean theorem can be used to find the true height ($h$) of a pyramid, which is the perpendicular distance from the apex to the center of the base. For a right pyramid with a square base of side length $b$ and a given slant height $s$, a right triangle is formed by the height ($h$), the slant height ($s$), and half the base length ($b/2$). The relationship is:

Examples

• A pyramid has a square base with a side of 10 meters and a slant height of 13 meters. What is its true height? We have $h^2 + (10/2)^2 = 13^2$, so $h^2 + 5^2 = 169$. This gives $h^2 = 169 - 25 = 144$, so the height is $h = 12$ meters.
• A pyramid's true height is 8 cm and its square base has a side of 12 cm. What is the slant height? The slant height $s$ is the hypotenuse. $s^2 = 8^2 + (12/2)^2 = 64 + 36 = 100$. So, the slant height is $s = \sqrt{100} = 10$ cm.
• To find the volume of a pyramid with a square base of 140 m and a slant height of 250 m, first find the height. $h^2 + (140/2)^2 = 250^2$, so $h^2 + 70^2 = 250^2$. This means $h^2 = 62500 - 4900 = 57600$, so $h=240$ m. The volume is $\frac{1}{3} \times (140^2) \times 240$ cubic meters.

Explanation

A pyramid's true height is hidden inside. You can find it by solving the right triangle formed by the slant height (the hypotenuse), the true height (a leg), and half the base's width (the other leg) using the Pythagorean theorem.

Property

For a rectangular pyramid with length $\ell$, width $w$, and height $h$, the volume is:

Examples