I know there are already many questions on this site about the area of the overlap between two circles, but I have a special case.
There is one circle with radius 1. There is another circle with unknown radius. The center of the circle with unknown radius is located on the circumference of the circle with radius 1.
Is there a formula to find the area of the overlap between the two circles?
I've tried looking at the generic formulas for area of overlap between two circles, but I've found they don't account for this edge case or they require the same perimeter.
Best Answer
Polar integration seems to be the best method for creating a general formula for this.
Re-expressing both circles as polar yields this: $r1 = 1$ and $r2 = cos\theta+\sqrt(a^2-sin^2\theta)$ based off of the general formula for a polar circle, centered at $(1, 0)$ with radius $a$. Really all we need here are the points of intersection which can easily be solved by setting each equation equal and following algebraic and trigonometric rules
$$cos\theta+\sqrt(a^2-sin^2\theta) = 1$$
$$a^2-sin^2\theta=1+cos^2\theta-2cos\theta$$
Replacing $sin^2$ and $cos^2$ with 1 and solving for theta yields:
$$\theta=\pm arcos(a^2-2/-2)$$
From here it should be noted that we must add $\pi$ to the resulting angle because our result accounts for the part of the circle outside of the central unit circle. So our angles from which we base our integral are
$$\theta = \pi \pm arcos(a^2-2/-2)$$
At this point I believe you can evaluate this regularly
$$.5\int_{\theta_1}^{\theta_2} 1^2-[cos\theta+\sqrt({a^2-sin^2\theta})]^2 \,d\theta$$
There may be some errors I missed in my answer however this general method should work