Question

The radii between the curved surface area and the total surface area of a right circular cylinder is 1:2. Prove that its height and radius are equal.

Answer 1

Let r be the radius and h be the height of a right circular cylinder, then
Curved surface area =$2\pi rh$
and total surface area =$2\pi rh\times 2\pi r^2=2\pi r(h+r)$
But their ratio is 1:2
$\therefore \frac{2\pi rh}{2\pi r(h+r)}=\frac{1}{2}\Rightarrow \frac{h}{h+r}=\frac{1}{2}$
$\Rightarrow 2h=h+r\Rightarrow 2h-h=r$
$\Rightarrow h=r$
Hence their radius and height are equal.