## Proof of Formula for the Curved Surface Area of a Frustum

The curved surface area of a cone of height
$h$
slant height
$l$
$r$
is
$A=\pi rl$
.
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
$\frac{L-l}{r}=\frac{L}{R} \rightarrow LR-lR=Lr \rightarrow L=\frac{lR}{R-r}$
and
$L-l=\frac{lR}{R-r}-l=\frac{lR-lR+lr}{R-r}=\frac{lr}{R-r}$
.
The surface area of the frustum is then
\begin{aligned} A_{FRUSTUM}&=\pi RL-\pi r(L-l) \\ &=\pi (R \frac{lR}{R-r} -r \frac{lr}{R-r} ) \\ &=\frac{\pi l}{(R-r)}(R^2-r^2) \\ &= \frac{\pi l}{(R-r)}(R-r)(R+r)= \pi l(R+r)\end{aligned}