The ratio between the curved surface area and the total surface area of a cylinder is 1 : 2. Find the ratio between the height and the radius of the cylinder.

It is given that the ratio between the curved surface area and the total surface area of a cylinder is 1 : 2.

$⇒ \dfrac{\text {Curved surface area of the cylinder}}{\text {Total surface area of the cylinder}} = \dfrac{1}{2}\\[1em] ⇒ \dfrac{2πrh}{2πr(r + h)} = \dfrac{1}{2}\\[1em] ⇒ \dfrac{\cancel{2πr}h}{\cancel{2πr}(r + h)} = \dfrac{1}{2}\\[1em] ⇒ \dfrac{h}{r + h} = \dfrac{1}{2}\\[1em] ⇒ 2 \times h = 1 \times (r + h)\\[1em] ⇒ 2h = r + h\\[1em] ⇒ 2h - h = r\\[1em] ⇒ h = r\\[1em] ⇒ \dfrac{h}{r} = \dfrac{1}{1}\\[1em]$

Hence, the ratio between the height and radius of the cylinder is 1 : 1.