Volume of Right Circular Cone

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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:


ConeVolumeProof.png


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:

$\overline {AC}$ as the axis of the cone, whose length is $h$
$A$ as the origin.

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