Question

A solid sphere and a solid hemi-sphere have the same total surface area.Find the ratio between their volumes.

Answer 1

surface of sphere = $4\pi r^2$

surface of hemisphere = $2\pi r^2+\pi r^2$

Equating these, with different radii, of course, we get

$4\pi r^2=3\pi R^2$ where r is radius of the sphere, R is the radius of hemisphere

r² =$\frac{3}{4}$R²

R = $\frac{2}{\sqrt{3}}$ r

$\frac{R}{r}$ = $\frac{2}{\sqrt{3}}$

Volumes are proportional to the cube of the above number, so Vol of hemisphere : vol of sphere = ($\frac{2}{\sqrt{3}}$ )³ = $\frac{8}{3\sqrt{3}}$