Square inscribed in a circle and an angle

Question

Consider this badly drawn picture and the circle in the picture. Suppose that the circle has unit length, so that the area is $\pi$. Suppose that We know that the area enclosed by $ABCD$ is exactly $\pi/12$. Can one then find the angle $CAB$?If so, how? I tried analytic methods, but to no real help.

Square in a circle

Hmm, a numerial approximation with Mathematica: 0.2681334894944452 — Alex, Jun 25 '14 at 15:06
Please crop the image link so that it doesn't have a full page of white space below the picture itself. — , Jun 25 '14 at 15:09

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

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Let $\theta$ be the angle $CAB$ in radians. Then the area ABCD is equal to $\frac{\theta+\sin(\theta)\cos(\theta)}{2}=\frac{\pi}{12}$. Rearranging that and using $\sin(2\theta)=2\sin(\theta)\cos(\theta)$, we get that $\theta$ satisfies $2\theta+\sin(2\theta)=\frac{\pi}{3}$.

I'm not sure if there exists an analytical solution to this: Solve $\sin x = 1 - x$

However, Mathematica gives $0.2681334894944452$ as a numerical approximation.

edited Apr 13 '17 at 12:21

Community

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answered Jun 25 '14 at 15:17

Alex

2,354

Where does the area formula come from? – rae306 Jun 25 '14 at 15:37
@RainiervanEs $\frac{\theta}{2}$ for the area $ABC$ and $\frac{\sin(\theta)\cos(\theta)}{2}$ for $ACD$. – Alex Jun 25 '14 at 15:39

Square inscribed in a circle and an angle

1 Answers1