$A=\int_{\alpha}^{\beta}\frac{1}{2}r^2d\theta$

where $\alpha$ is the starting angle and $\beta$ is the ending angle.

To find the area that is enclosed by two polar equations like in the picture below, we use the formula:

$A=\int_{\alpha}^{\beta}\frac{1}{2}(r^{2}\;_{outer}-r^{2}\;_{inner})d\theta$

where $r_{outer}$ is the outer part of the first polar equation, and $r_{inner}$ is the inner part of the second polar equation.