## Proof of Formula for Volume of Frustum

The volume of a cone of height
$h$
and base radius
$r$
is
$V=\pi r^2 h$
.
A frustum is a truncated cone. Part of the top is cut off by a cut parallel to the base.

The whole cone and the top section are similar cones, so
$\frac{H-h}{r}=\frac{H}{R} \rightarrow HR-hR=Hr \rightarrow H=\frac{hR}{R-r}$
and
$H-h=\frac{hR}{R-r}-h=\frac{hR-hR+hr}{R-r}=\frac{hr}{R-r}$
.
The volume of the frustum is then
\begin{aligned} V_{FRUSTUM}&=\frac{1}{3}\pi R^2H-\frac{1}{3}\pi r^2(H-h) \\ &=\frac{1}{3}\pi(R^2H-r^2(H-h)) \\ &=\frac{1}{3}\pi(R^2(\frac{hR}{R-r})-r^2 (\frac{hr}{R-r})) \\ &=\frac{1}{3(R-r)}\pi (R^3-r^3) \end{aligned}

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