In a right circular cylinder, the ratio of the curved surface area to the total surface area is 2 : 5. Find the ratio of the height to the radius of the cylinder.

The ratio of the curved surface area (CSA) to the total surface area (TSA) of the cylinder is 2:5.

Let the radius of the cylinder be r and the height be h.

Curved Surface Area (CSA) of a cylinder = $$2\pi r h$$​​

Total Surface Area (TSA) of a cylinder = $$2\pi r h + 2\pi r^2$$​​

​$$\frac{\text{CSA}}{\text{TSA}} = \frac{2}{5}\\ \\\frac{2\pi r h}{2\pi r h + 2\pi r^2} = \frac{2}{5} \\\frac{h}{h + r} = \frac{2}{5}\\5h = 2(h + r)\\5h = 2h + 2r\\5h – 2h = 2r\\3h = 2r \\$$​​

Now,

​$$\frac{h}{r} = \frac{2}{3}$$​​

Thus, The ratio of the height to the radius of the cylinder is 2:3.