Q. 23

Find the ratio of the volume of a cube to that of a sphere which will fit inside it.

Answer :

Let the radius of the sphere be ‘R’ units


And the cube which will fit inside it be of edge ‘a’ units


Explanation: The longest diagonal of the cube that will fit inside the sphere will be the diameter of the of the sphere.


The longest diagonal of cube = the diameter of the sphere


Consider ΔBCD, BDC = 90°


BD = CD = a units (as they are the edges of cube)



BC2 = a2 + a2 (putting value of BD and CD)


BC2 = 2a2


BC = √(2a2)


BC = a√2 units eqn1


Now consider ΔABC, ABC = 90°


Here, AB = a units and BC = a√2 units



AC2 = a2 + (a√2)2 (putting values of AB and BC)


AC2 = a2 + 2a2


AC2 = 3a2


AC = √(3a2)


AC = a√3 units


Diameter of sphere = D = a√3 units


And we know, D = 2 × R


R = D/2 (put value of D )



Also, Volume of a sphereeqn2


Put value of R in eqn2




Volume of sphere = πa2 cubic units eqn3


Volume of cube = (edge)3


Volume of cube = a3 cubic units eqn4


Ratio of volume of cube to that of sphere


(putting values from eqn3 and eqn4)


Ratio of volume of cube to that of sphere



Ratio of volume of cube to that of sphere is a:π


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