Volume of a Sphere
If a sphere is placed in a cylinder of the same diameter and with an height equal to the diameter and the vacant spaces are filled with sand, the sand will be found to fill exactly onethird of the cylinder after the sphere is removed.
Therefore, a sphere is equal to twothirds of a cylinder of the same diameter and height.
Here the volume of the cylinder $$ = 2\pi {r^2} \times 2r = 2\pi {r^3}$$
$$\therefore $$ The volume of the sphere $$ = \frac{2}{3} \times 2\pi {r^3} = \frac{4}{3}\pi {r^3}$$
Summary:

The volume of a sphere $$ = \frac{4}{3}\pi {r^3}$$ or $$\frac{\pi }{6}{d^3}$$, where $$r$$ is the radius and $$d$$ is the diameter of the sphere.

The volume of a hemisphere $$ = \frac{2}{3}\pi {r^3}$$