Fraction of circle contained in another circle passing through its center.

Question

Consider a circle $C$ with radius $r$. Now take any point on the boundary of the circle, say $P$, and draw another circle $C'$ with $P$ as the centre and radius $k\cdot r (0\le k \le 2)$. Now what is the ratio of the intersection area of $C$ and $C'$ to the area of $C'$? Note that in the limiting case of $k=2, C'$ "engulfs" $C$ and thus the ratio is $.25$. But I need a general answer in terms of $k$.

Answer 1

Let's assume that we have a unit circle, and that the point $P$ is on the $x$-axis, i.e, it's the point $(0, 1)$. Then the intersection area will be symmetric about the $x$ axis, so let's just do this for the upper half of the intersection area.

The new circle intersects the unit circle at some point $S$ which is $(\cos u, \sin u)$ for some number $u$. I'm going to hope that you can relate $k$ to $u$ by yourself, and just work from $u$. I'll call the origin of the unit circle $T$.

To find the area of the "sorta triangle" intersection, I'm going to write it as a sum of the area of the wedge -- the chunk of the unit circle from the origin $T$ to $P = (1, 0)$ to $S$ -- and the area of the "sliver" --- the stuff whose angular polar coordinate is greater than $u$.

The wedge has area $$ A_1 = u, $$ by elementary geometry.

The sliver is trickier...but look at it from the point of view of polar coords around $P$: it's just a wedge of the smaller circle, minus the area of the "straight-line triangle $TPS$" contained in it.

The area of $TPS$ is $$ A_2 = \frac{1}{2} \sin(u) \cos(u). $$ The wedge of the smaller circle...depends on the angle at $P$. That angle is $$ v = \arcsin(\sin(u)/k), $$ so the area of that wedge is $$ A_3 = v k^2 $$

And the total is $A_1 + A_3 - A_2$. We divide this by the area of the upper half of the small circle -- $\pi k^2/2$ -- to get the answer:

$$ ratio = \frac{u + vk^2 - \frac{1}{2} \sin(u) \cos(u)}{\frac{\pi k^2}{2}}\\ = \frac{2u + 2vk^2 - \sin(u) \cos(u)}{\pi k^2}, $$ where

$$ v = \arcsin(\sin(u)/k). $$