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. h1 = h2 = h3 = h & r1 = r2 = r3 = r
As we know that,
Surface Area of Sphere : 4 π r12
Curved Surface Area of cylinder: 2 π r2 h2
Curved Surface Area of cone: π r3 l –> where l is slant height, l = sqrt(r32 + h3 2 )
ratio of curved surfaces is:
-> 4 π r12 : 2 π r2 h2 : π r3 sqrt(r32 + h3 2 )
as per condition given in the question we can show that,
Height of sphere, h1 = 2 x radius = 2 r1 , i.e. h = 2r
so, ratios can be simplified as
-> 4 π r2 : 2 π r (2r) : π r sqrt(r2 + (2r) 2 )
-> 4 r : 2 (2r) : sqrt(5r2 )
-> 4 r : 4 r : r sqrt(5)
-> 4 : 4 : sqrt(5) # final answer
Also remember:
Volume of Sphere: (4/3) π r3
Volume of cylinder: π r2 h
Volume of cone: (1/3) π r2 h