## Volume of a Cone

A cone is a three-dimensional figure with one circular base. A curved surface connects the base and the vertex.

The **volume** of a 3 -dimensional solid is the amount of space it occupies. Volume is measured in cubic units (** in**^{3},ft^{3},cm^{3},m^{3}, et cetera). Be sure that all of the measurements are in the same unit before computing the volume.

The volume V of a cone with radius r is one-third the area of the base B times the height h .

$V=\frac{1}{3}Bh\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}V=\frac{1}{3}\pi {r}^{2}h,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{where}\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=\pi {r}^{2}$

**Note :** The formula for the volume of an oblique cone is the same as that of a right one.

The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism are related. If the heights of a cone and a cylinder are equal, then the volume of the cylinder is three times as much as the volume of a cone.

**Example:**
Find the volume of the cone shown. Round to the nearest tenth of a cubic centimeter.

**Solution:**

From the figure, the radius of the cone is 8 cm and the height is 18 cm.

The formula for the volume of a cone is,

$V=\frac{1}{3}\pi {r}^{2}h$

Substitute 8 for $r$ and 18 for h .

$V=\frac{1}{3}\pi {\left(8\right)}^{2}\left(18\right)$

Simplify.

$\begin{array}{l}V=\frac{1}{3}\pi \left(64\right)\left(18\right)\\ =384\pi \\ \approx 1206.4\end{array}$

Therefore, the volume of the cone is about 1206.4 cubic centimeters.