Lesson 2: Volumes of Cones

Property

A cone is a three-dimensional shape with a circular base and an apex. The volume of a right circular cone is one-third the product of the area of the base and the height. If the height is $h$ and the radius of the base is $r$, then $V = \frac{1}{3} \pi r^2 h$.

Examples

• A party hat is a cone with a height of 8 inches and a base radius of 3 inches. Its volume is $V = \frac{1}{3} \pi (3^2)(8) = 24\pi$ cubic inches.
• A cylinder has a volume of $90$ cubic cm. A cone with the same base and height would have a volume of $V = \frac{1}{3} (90) = 30$ cubic cm.
• An hourglass cone holds $15\pi$ cubic inches of sand and has a base radius of 3 inches. Its height is found by solving $15\pi = \frac{1}{3} \pi (3^2)h$, which gives $h=5$ inches.

Explanation

A cone's volume is directly related to a cylinder's. For a cone and cylinder with the same base radius and height, the cone's volume is exactly one-third of the cylinder's. You could fit three full cones of water into one cylinder.

Property

To find the height of a cone when given its volume and radius, solve the cone volume formula $V = \frac{1}{3}\pi r^2 h$ for $h$:

1. Start with the cone volume formula: $V = \frac{1}{3}\pi r^2 h$
2. Multiply both sides by 3: $3V = \pi r^2 h$
3. Divide both sides by $\pi r^2$: $h = \frac{3V}{\pi r^2}$

Examples

Property

When the volume and height of a cone are known, you can find the radius by rearranging the cone volume formula $V = \frac{1}{3}\pi r^2 h$. Isolate $r^2$ by multiplying both sides by 3 and dividing by $\pi h$:

Since radius is a physical dimension, we use only the positive square root.

Examples