Question

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

• A. $4:4:\sqrt{5}$
• B. $4:5:\sqrt{5}$
• C. $3:4:\sqrt{5}$
• D. $2:3:2\sqrt{5}$

Answer 1

The correct option is A $4:4:\sqrt{5}$

Since, the height of a sphere is the diameter, the cone and cylinder have height 2r.

Then

Curved surface area of Sphere=$4\pi r²$

Curved surface area of cylinder =$2\pi r\left(2r\right)=4\pi r²$

Curved surface area of cone = $\pi rl$

where, $l=\sqrt{(r^2+h^2})=\sqrt{(r2+(2r{)}^2})=\sqrt{5r^2}=r\sqrt{5}$

⇒ Curved surface area of cone =$\pi \sqrt{5r^2}$

Now,

ratio of CSA 's a sphere ,cylinder and a cone = $4\pi r^2:2\pi rh:\pi rl$

=$4\pi r^2:4\pi r^2:\pi r^2\sqrt{5}$

=$4:4:\sqrt{5}$