Surface of the solid generated by a right triangle about its hypotenuse

Text

A right triangle has legs in ratio $3:4$ and hypotenuse $25$ cm. It makes a full turn about its hypotenuse. Find the surface area of the resulting solid.

Solution

Legs $3k,4k$ with $5k=25\Rightarrow k=5$, i.e. $15$ cm and $20$ cm. Rotation about the hypotenuse generates two cones sharing a base of radius equal to the altitude to the hypotenuse: $\rho=\frac{15\cdot20}{25}=12\ \text{cm}.$ The slant heights of the two cones are the legs $15$ and $20$. The surface (the two lateral surfaces) is $S=\pi\rho(15+20)=\pi\cdot12\cdot35=420\pi\approx1319.47\ \text{cm}^2.$ $\boxed{S=420\pi\approx1319.47\ \text{cm}^2}$