Step 1: Understand the given question and make a note of the given information

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

Let r be the common radius of a sphere, a cone and cylinder.

Height of sphere = its diameter = 2r

Step 2: Calculate the curved surface areas of the sphere, cylinder and cone

Curved surface area of Sphere = 4 \pi r^2

Curved surface area of cylinder = 2 \pi r h

= 2 \pi r (2r)

= 4 \pi r^2

Curved surface area of cone = \pi r l

Slant height of the cone l = \sqrt{r^2 + h^2}

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

l = \sqrt{5}r

Curved surface area of cone = \pi r \sqrt{5} r

= \sqrt{5} \pi r^2

Step 3: Find the ratio of curved surface areas

sphere:cylinder:cone = 4 \pi r^2:4 \pi r^2:\sqrt{5} \pi r^2

= 4:4:\sqrt{5}