# How do I use polar coordinates to find the volume of a sphere of radius r?

The volume is $V = \frac{4}{3} \pi {r}^{3}$

#### Explanation:

The equation of a sphere is ${x}^{2} + {y}^{2} + {z}^{2} = {r}^{2}$

From the equation we get

z=+-sqrt(r^2-(x^2+y^2)

The volume of the sphere is given by

$V = 2 \int {\int}_{{x}^{2} + {y}^{2} \le r} \sqrt{{r}^{2} - {x}^{2} - {y}^{2}} \mathrm{dA}$

Using polar coordinates $x = r \cos a , y = r \sin a$ and substituing

to the integral above

$V = 2 {\int}_{0}^{2 \cdot \pi} {\int}_{0}^{r} \sqrt{{r}^{2} - {a}^{2}} r \mathrm{dr} \mathrm{da}$

Which is calculated easily giving $V = \frac{4}{3} \pi {r}^{3}$