Volume of Right Circular Cone

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 length between the apex and the center of the base, that is, the cone's height.

Construction
Construct the following triangle:


 * ConeVolumeProof.png

We can create any right circular cone by rotating this plane region about the $x$-axis.

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

Let $AB$ be a line segment with constant slope $m$.

Let $AC$ be the axis of the cone.

By definition, its length is the height $h$ of the cone.

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:


 * $\displaystyle (1): \quad V = \pi \int_A^C \left({R \left({x}\right)}\right)^2 \ \mathrm d x$

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 $AB$, and so:


 * $R \left({x}\right) = m x$

We have also defined:
 * $AC$ as the axis of the cone, whose length is $h$
 * $A$ as the origin.

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

By construction, the base of the cone has a radius $r$ given by:


 * $r = R \left({h}\right) = mh$

Substituting $r$ for $mh$:


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