How do I find the perimeter of these overlapping circles?

The correct answer is 99.5

  • 2
    $\begingroup$ What are your thoughts? What did you try? It's not about answer but about how you think. $\endgroup$ – yW0K5o May 13 at 14:56
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    $\begingroup$ Look at the intersection points of the two circles. Can you find the angle between these two points, when the vertex of the angle is at the center of one of the circles? $\endgroup$ – Andrei May 13 at 15:03
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    $\begingroup$ Observe that the triangle formed by the two centers and one of the points of intersection of the two circles has sides $9,12,15$, which satisfies Pythagoras. Therefore, it is a right angle triangle. Therefore, the angles are $\pi/2, \arctan(4/3)$ and $\arctan(3/4)$. The double of the last two multiplied by the corresponding radii give you the lengths of those arcs that you need to subtract $\endgroup$ – logarithm May 13 at 15:04
  • $\begingroup$ I got $99.8...$ $\endgroup$ – J. W. Tanner May 13 at 15:27

Denote the points of intersection of the circles as $A$ and $B$ and the centers as $O_1$ and $O_2$.

In triangle $O_1AO_2$ you have three sides so you can find the angles. Then from the information about the angles and the radii of circles you can find the arc lengths and the perimeter.


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