# Ratio of Curved Surfaces of Sphere, Cylinder and Cone

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

- 4: √5 :4
- 2: 2: √5
- 4: 4: 5
- 4: 4: √5

**Answer:** 4 : 4 : √5

**Solution:**

It is provided in question that radius & height of sphere, cylinder and cone are same.

i.e. h_{1 }= h_{2 }= h_{3 } = h & r_{1} = r_{2} = r_{3} = r

As we know that,

Surface Area of Sphere : 4 π r_{1}^{2}

Curved Surface Area of cylinder: 2 π r_{2} h_{2}

Curved Surface Area of cone: π r_{3} l –> where l is slant height, l = sqrt(r_{3}^{2} + h_{3 } ^{2 })

ratio of curved surfaces is:

-> 4 π r_{1}^{2} : 2 π r_{2} h_{2} : π r_{3} sqrt(r_{3}^{2} + h_{3 } ^{2 })

as per condition given in the question we can show that,

Height of sphere, h_{1} = 2 x radius = 2 r_{1 } , i.e. h = 2r

so, ratios can be simplified as

-> 4 π r^{2} : 2 π r (2r) : π r sqrt(r^{2} + (2r) ^{2 })

-> 4 r : 2 (2r) : sqrt(5r^{2} )

-> 4 r : 4 r : r sqrt(5)

**-> 4 : 4 : sqrt(5) ** # final answer

**Also remember:**

Volume of Sphere: (4/3) π r^{3}

Volume of cylinder: π r^{2} h_{ }Volume of cone: (1/3) π r^{2} h