Question

A sphere, a cylinder and a cone are of the same radius and same height. Find the ratio of their curved surfaces.

Answer 1

Given:

A sphere, a cylinder and a cone have the same radius and the same height.

Find the ratio of their curved (lateral) surfaces.

Let the common radius = r and common height = h. For a sphere the height = diameter = 2r, so h = 2r.

The curved surface of the sphere = total surface area = `4πr^2.`

Curved surface of the cylinder = 2πr h = 2πr(2r) = 4πr².

`= sqrt(r^2 + h^2) = sqrt(r^2 + (2r)^2) = rsqrt5.`

Curved surface of the cone = `π r l = π r (rsqrt5) = π r^2 sqrt5.`

Ratio (sphere : cylinder : cone) = `4πr^2 : 4πr^2 : π r^2 sqrt5`

Divide through by π r^2 to simplify ⇒ 4 : 4 : `sqrt5.`