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Volume of a Cone

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

Math diagram

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 $h$ .

$V = \frac{1}{3} B h or V = \frac{1}{3} π r^{2} h, where B = π r^{2}$

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.

Math diagram

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,

$V = \frac{1}{3} π r^{2} h$

Substitute $8$ for $r$ and $18$ for $h$ .

$V = \frac{1}{3} π {(8)}^{2} (18)$

Simplify.

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

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