Question

If a sphere is inscribed in a cube, then the ratio of the volume of the sphere to the volume of the cube is
(a) $\pi :2$
(b) $\pi :3$
(c) $\pi :4$
(d) $\pi :6$

Answer 1

The correct option is (d): $\pi :6$

In the given problem, we are given a sphere inscribed in a cube. This means that the diameter of the sphere will be equal to the side of the cube. Let us take the diameter as $d.$

therefore Diameter of sphere, $d=$ Side of the cube $\left(s\right)$

Here, The volume of a sphere $\left(V_1\right)=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (\frac{d}{2}{)}^3$

$=\left(\frac{4}{3}\right)\pi \left(\frac{d^3}{8}\right)$

$=\frac{\pi d^3}{6}$

Volume of a cube $\left(V_2\right)=s^3=d^3$ [$\because d=s$]

Now, the ratio of the volume of the sphere to the volume of the cube $=\frac{V_1}{V_2}$

$\frac{V_1}{V_2}=\frac{\frac{\pi d^3}{6}}{d^3}=\frac{\pi }{6}$

So, the ratio of the volume of the sphere to the volume of the cube is $\pi :6.$ Therefore,