Solve pirsqrt{h^2+r^2}=s

$$\pi r \sqrt { h ^ { 2 } + r ^ { 2 } } = s$$

Solve for h

$$\left\{\begin{matrix}h=\frac{\sqrt{s^{2}-\pi ^{2}r^{4}}}{\pi |r|}\text{; }h=-\frac{\sqrt{s^{2}-\pi ^{2}r^{4}}}{\pi |r|}\text{, }&\left(r\leq \frac{\sqrt{s}}{\sqrt{\pi }}\text{ and }s\geq 0\text{ and }r>0\right)\text{ or }\left(r\geq -\frac{\sqrt{-s}}{\sqrt{\pi }}\text{ and }s\leq 0\text{ and }r<0\right)\\h\in \mathrm{R}\text{, }&s=0\text{ and }r=0\end{matrix}\right.$$

Solve for r

$$\left\{\begin{matrix}r=-\frac{\sqrt{\frac{2\sqrt{4s^{2}+\pi ^{2}h^{4}}}{\pi }-2h^{2}}}{2}\text{, }&s\leq 0\\r=\frac{\sqrt{\frac{2\sqrt{4s^{2}+\pi ^{2}h^{4}}}{\pi }-2h^{2}}}{2}\text{, }&s\geq 0\end{matrix}\right.$$