Here, $$ CSA of cone $=\mathit{\pi }\mathit{R}\mathit{l}$
Radius of sphere $=\mathit{r}$
height of cone $=\mathit{h}$
In $△\mathrm{AOC}$ ,
$\mathit{A}{\mathit{O}}^2=\mathit{A}{\mathit{C}}^2+\mathit{O}{\mathit{C}}^2$
$\Rightarrow {\mathit{r}}^2={\mathit{R}}^2+{\left(\mathit{h}-\mathit{r}\right)}^2$
$\Rightarrow {\mathit{R}}^2=2\mathit{h}\mathit{r}-{\mathit{h}}^2$
$\therefore$ Radius of cone, $\mathit{R}=\sqrt{2\mathit{h}\mathit{r}-{\mathit{h}}^2}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(\mathit{i}\right)$
In $△\mathrm{ABC}$ ,
$\mathit{A}{\mathit{B}}^2=\mathit{A}{\mathit{C}}^2+\mathit{B}{\mathit{C}}^2$
$\Rightarrow {\mathit{l}}^2={\mathit{R}}^2+{\mathit{h}}^2$
$\Rightarrow {\mathit{l}}^2=2\mathit{h}\mathit{r}-{\mathit{h}}^2+{\mathit{h}}^2$
∴ slant height $=\sqrt{2\mathit{h}\mathit{r}}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(\mathit{i}\mathit{i}\right)$
CSA of cone $=\mathit{\pi }\mathit{R}\mathit{l}$
$=\mathit{\pi }\sqrt{2\mathit{h}\mathit{r}-{\mathit{h}}^2}\sqrt{2\mathit{h}\mathit{r}}$

$=2{\mathit{\pi }}^2\mathit{h}\mathit{r}\left(2\mathit{h}\mathit{r}-{\mathit{h}}^2\right)$
$=2{\mathit{\pi }}^2\mathit{r}\left(2\mathit{r}{\mathit{h}}^2-{\mathit{h}}^3\right)$