Derive/proof volume of Spherical sector

Question

How should I derive/proof volume of Spherical sector ($V=\frac{2 \pi r^2h}{3}$) without using integration, but knowing formulas for volume of cylinder, sphere and cone?

Answer 1

Consider how much of the surface area is covered by the sector as a fraction of the whole. The surface area is given by $2 \pi r h$, since this is the area that the cylinder enclosing it would have. So, the sector covers $\frac{h}{2r}$ of the sphere, and hence has volume $\frac{h}{2r} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^2 h$.