# How to find the area of the shaded region? [closed]

How do i find the area of the following shaded region?

The figure consists of two circles, one of radius $2r$ and the other of radius $r$.

The distance of the center of the circle of radius $r$ from the bigger circle is $r$, as shown.

The total angle is $\theta$---$\left(\dfrac{\theta}{2}+\dfrac{\theta}{2}\right)$

## closed as off-topic by pkwssis, Gina, Adam Hughes, Cookie, WilliamJul 25 '14 at 3:59

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – pkwssis, Gina, Adam Hughes, Cookie, William
If this question can be reworded to fit the rules in the help center, please edit the question.

If we take a Cartesian reference in the center of the small circle, so that $$x^2+y^2=r^2\\ (x+r)^2+y^2=(2r)^2$$ are the equations of the two circles, then in polar coordinates the two equations reduce to $$\rho=r\\ \rho=r\left(\sqrt{3+\cos^2\phi}-\cos\phi\right)$$ The requested area is given by $$A=\int_{-\theta/2}^{+\theta/2}d\phi\int_r^{r\left(\sqrt{3+\cos^2\phi}-\cos\phi\right)}\rho d\rho$$
Represent the shaded region as an algebraic sum ($+$- and $-$- signs) of circular sectors and triangles.