The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2.
Question:

The ratio 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.

Solution:

Let $S_{1}$ and $S_{2}$ be the curved surface area and total surface area of the circular cylinder, respectively.

Then, $S_{1}=2 \pi r h, \mathrm{~S}_{2}=2 \pi r(r+h)$

According to the question:

$S_{1}: S_{2}=1: 2$

$2 \pi r h: 2 \pi r(r+h)=1: 2$

$h:(r+h)=1: 2$

$\frac{h}{r+h}=\frac{1}{2}$

$2 h=r+h$

$h=r$

Therefore, the height and the radiu $s$ are equal.