What is the area of the cusp between two equi-radius circles?

Question

I have two circles that have the same radius $r$ where the centers are a distance apart $d < 2r$. How would I go about finding the area highlighted in yellow? What is that region called?

Answer 1

You can use calculus; suppose origin of coordinates is at midpoint of d, so the coordinates of center of right circle is $(\frac d2, 0)$. The equation of tangent to circles is $y=r$ and that of circle is:

$(x-\frac d2)^2+y^2=r^2 \Rightarrow y=\sqrt{r^2-(x-\frac d2)^2}$

So the area between $(\frac d2, 0)$ is:

$a=\int^{\frac d2}_0 [r-\sqrt {r^2-(x-\frac d2)^2}]dx$

This is for upper part, if you multiply it by 2 you get total colored area.

Let $x-\frac d2=X$ we have:

$d(x-\frac d2)^2=2xdx=2XdX\Rightarrow dX=dx$

$\int \sqrt{r^2-X^2} dX=\frac X2 \sqrt {r^2-X^2}+\frac{r^2}2 \sin^{-1}\frac X r$

putting intervals in we finally get:

$A=2a=r\cdot d+\frac d2 \sqrt {r^2-\frac {d^2}4}+ r^2 \sin^{-1} \frac d{2r}$