I have two circles which share a radius of R units, and each circle contains the center of the other circle. I found that the area of the segment would be, $\theta$ is the central angle between the two radii.

$A=\frac{1}{2}R^2 (\theta - \sin(\theta)).$

I have to explain why $\theta = 2\pi/3$

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    $\begingroup$ Where is the "upper diagram"? $\endgroup$ – 11684 May 7 '15 at 10:55
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    $\begingroup$ What does "share a radius" mean? $\endgroup$ – mvw May 7 '15 at 10:56
  • $\begingroup$ link the arc of each circle is placed on the each others radius $\endgroup$ – Caroline Adams May 7 '15 at 10:57

Set up the circles so that their centers are on a horizontal line, one to the left and one to the right, and the centers $A$ and $B$ are distance $R$ apart, with $A$ being the left one. There are two intersection points between the circles, an upper, $U$, and a lower, $V$.

Then the distance from $A$ to $U$ is $R$, as is the distance from $B$ to $U$. And the distance from $A$ to $B$ is $R$ as the left circle has radius $R$ and the point $B$ is on it. Hence the triangle $AUB$ is equilateral and has all angles $\pi/3$. The same goes for $AVB$, so the angles at $A$ and at $B$ (namely $VAU$ and $VBU$ must be $2\pi/3$.

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    $\begingroup$ I understand the answer. On the other hand, I still don't understand the question. $\endgroup$ – zoli May 7 '15 at 11:14
  • $\begingroup$ The two circles I've shown "share a radius of R units" in that both have radius $R$. The left circle contains the right one's center. "The two radii" are $AU$ and $AV$, and "it" refers to the diamond shape. How do I know this? Once I'd drawn the two circles and looked for something with area $\frac{R^2}{2}(\theta - \sin \theta)$, I figured out what $\theta$ and "it" had to be. :) $\endgroup$ – John Hughes May 7 '15 at 11:20

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