Ratio of Curved Surfaces of Sphere, Cylinder and Cone

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