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Two circles with centers $A$ and $B$ of radius $10$ intersect at points $B$ and $C$ such that $AB = 16$. $\angle BAC=\angle ABC = 0.64\,\operatorname{rad}$ and $\angle ACB = 1.86\,\operatorname{rad}$. The arc length $CD = 12.87$.

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I need to find the area of the shaded region and I can't think of any way to do it.

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    $\begingroup$ Hint: $CD=12$ by Pythagoras theorem. $\endgroup$ – Tunk-Fey May 26 '14 at 17:45
  • $\begingroup$ See this Area of intersection between two circles. It can help you. $\endgroup$ – Jika May 26 '14 at 17:48
  • $\begingroup$ @Jika In that question, the circles are a radius distance apart. $\endgroup$ – usama8800 May 26 '14 at 17:53
  • $\begingroup$ Yes but it is simply a hint. It can give you a way of thinking to solve your problem. @usama8800 $\endgroup$ – Jika May 26 '14 at 18:00
  • $\begingroup$ @Tunk-Fey thanks, I got it. $\endgroup$ – usama8800 May 26 '14 at 18:06
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If you draw the line segment from $C$ to $D$, it divides the shaded region into two pieces called circular segments. One of the circular segments is a segment of the circle with center at $A$, and the other is a segment of the circle with center at $B$.

You can see here -- http://mathworld.wolfram.com/CircularSegment.html -- how to compute the area of a circular segment.

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