Question

The sphere is centered in the cube. Find an expression for the volume of the cube outside the sphere

Answer 1

Solution -

The sphere is centered in the cube, so diameter of the sphere is equal to the side of the cube. Hence ,the radius of the sphere is half of the length of the side.

If the length of side of the cibe = a , then radius of the sphere is $\frac{a}{2}$

Volume of the cube = a³, volume of the sphere = $\frac{4}{3}$π(radius)³ = $\frac{4}{3}$π($\frac{a}{2}$)³

volume of the cube outside the sphere = volume of the cube - volume of the sphere

=  $a^3 - \frac{4}{3}$π($\frac{a}{2}$)³

= $a^3 -\frac{1}{6} \pi a^{3}$

=a³(1 - $\frac{\pi }{6}$)