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

  1. 4: √5 :4
  2. 2: 2: √5
  3. 4: 4: 5
  4. 4: 4: √5
Manish Listener Asked on 15th June 2015 in Maths.
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1 Answer(s)

Answer: 4 : 4 : √5

Solution:

It is provided in question that radius & height of sphere, cylinder and cone are same.
i.e.  h= h2 = h = 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 r , 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

Pushpendra Pal Dean Answered on 31st October 2014.
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