Proof of Formula for the Curved Surface Area of a Frustum

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

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