Question

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

(a) 3 : 2 : 1 

(b) 4 : √3 : 4 

(c) 4 : 4 : √5 

(d) 3: 3 : √5

Answer 1

(c) 4 : 4 : √5

Let r be the common radius of the sphere, cone and cylinder. Then, Height of the cone = Height of the cylinder = Height of the sphere = 2r 

Let l be the slant height of the cone. Then

\(l=\sqrt{h^2+r^2}=\sqrt{4r^2+r^2}=r\sqrt5\)

Now, CSA of sphere = 4πr² 

CSA of cylinder = 2πr. 2r = 4πr² 

CSA of cone = πrl = πr. \(\sqrt5πr^2 = \sqrtπr^2\)

∴ Required ratio = 4πr² : 4πr² : \(\sqrt5\,πr^2 \) = 4 : 4 : √5