If the radius of a cone is \(r\) units, the height is \(h\) units, and the curved surface area is \(S\) square units, then prove that \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

Q.If the radius of a cone is \(r\) units, the height is \(h\) units, and the curved surface area is \(S\) square units, then prove that \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

\(\therefore\) Slant height of the cone = \(\sqrt{r^2 + h^2}\) units \(\therefore\) Curved surface area of the cone, \(\pi r \sqrt{r^2 + h^2} = S\) i.e., \(\pi^2 r^2 (r^2 + h^2) = S^2\) i.e., \(\pi^2 r^4 + \pi^2 r^2 h^2 = S^2\) i.e., \(\pi^2 r^2 h^2 = S^2 - \pi^2 r^4\) i.e., \(\pi r h = \sqrt{S^2 - \pi^2 r^4}\) i.e., \(h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r}\) (Proved)