# Volume of Right Circular Cone

## Contents

## Theorem

The volume $V$ of a right circular cone is given by:

- $V = \dfrac 1 3 \pi r^2 h$

where:

- $r$ is the radius of the base
- $h$ is the height of the cone, that is, the distance between the apex and the center of the base.

## Construction

Construct the following triangle:

Let $A$ be located at the origin of the $xy$-plane.

By definition of tangent, the line segment $\overline {AB}$ can be described by the equation $y = \dfrac {BC} {AC} x$.

By Euclid's definition of a cone, the solid of revolution generated by rotating $\triangle ABC$ about the $x$-axis is a right circular cone:

- whose axis is $\overline {AC}$
- whose base consists of the circle whose center is $C$, whose radius is $BC$ and whose plane is perpendicular to $\overline {AC}$.

As $\overline {AC}$ is perpendicular to the base of the cone, the height of the cone is $AC$.

Let $h = AC$ denote the height and $r = BC$ denote the radius of the base.

## Proof

This proof utilizes the Method of Disks and thus is dependent on Volume of a Cylinder.

From the Method of Disks, the volume of the cone can be found by the definite integral:

- $\displaystyle (1): \quad V = \pi \int_0^{AC} \left({R \left({x}\right)}\right)^2 \ \mathrm d x$

where $R \left({x}\right)$ is the function describing the line which is to be rotated about the $x$-axis in order to create the required solid of revolution.

In this example, $R \left({x}\right)$ describes the line segment $\overline {AB}$, and so:

- $R \left({x}\right) = \dfrac r h x$

We have also defined:

So the equation $(1)$ is interpreted as:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle V\) | \(=\) | \(\displaystyle \) | \(\displaystyle \pi \int_0^h \left({\frac r h x}\right)^2 \ \mathrm d x\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle \left.{\pi \left({\frac r h}\right)^2 \frac {x^3} 3}\right \vert_{x=0}^{x=h}\) | \(\displaystyle \) | \(\displaystyle \) | Constant Multiple Rule, Power Rule | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle \frac 1 3 \pi r^2 h\) | \(\displaystyle \) | \(\displaystyle \) |

$\blacksquare$

## Sources

- Weisstein, Eric W. "Cone." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Cone.html - Weisstein, Eric W. "Method of Disks." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/MethodofDisks.html