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 $CD = 12.87$.


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

  • 2
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.