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There are two circles $C_1$ and $C_2$. The radius of the circle $C_1 = r$, and area of $C_1 = s$. The center of circle $C_2$ lies on the border of circle $C_1$. The area of the intersection of the two circles is $s/2$.

What is the radius of circle $C_2$?

user3209698
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Arthurghev
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1 Answers1

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This is known as the Goat Problem.

For simplicity we could set $r=1$ and $s=\pi r^2=\pi$, and assume the radius of $C_2$ is $R$. Using the formula in http://mathworld.wolfram.com/Circle-CircleIntersection.html:

\begin{align} \text{area of intersection} &= r^2\cos^{-1}\frac{d^2+r^2-R^2}{2dr} + R^2\cos^{-1}\frac{d^2+R^2-r^2}{2dR} \\ &\quad - \frac{\sqrt{(-d+r+R)(d+r-R)(d-r+R)(d+r+R)}}2 \end{align}

with $r=1$ and $d=1$ (since the center of $C_2$ is on the circumference of $C_1$) we get to solve

$$ R^2 \cos^{-1} \frac R2 + \cos^{-1}\left(1-\frac{R^2}2\right) - \frac{R\sqrt{4-R^2}}2 = \frac\pi2 $$ which doesn't have a closed-form solution. The numerical solution is $$ R=1.15872847301812151782823355653641924698\dotsc \times r.$$

kennytm
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