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There is a circular pen with a goat in it.

The goat is tethered by a rope to the edge of the pen.

The rope is the length of the radius of the pen.

What area of grass in the pen can the goat graze?

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    $\begingroup$ Did you try and draw a picture? $\endgroup$ – Pedro Tamaroff Jun 6 '13 at 22:24
  • $\begingroup$ Yes I did. It didn't help much. $\endgroup$ – Brad Jun 6 '13 at 22:27
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enter image description here

Note that the area is the darker shaded red one. Argue that the area of that is four times the area of the shape $FBD$. Then observe that this area is precisely the are of the circular sector $ABD$ minus the area of the triangle $AFD$. Find the angle $\angle DAF$ and using that the circle has radius $r$, find the area of $ADBC$.

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Hint: you have a lens shaped area that is the intersection of two circles. If you draw the linesegment connecting the intersection points of the two circles, you get two circular segments

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If I may "piggy-back" on Peter Tamaroff's diagram, it will be useful to consider the fact that the lens-shaped area where the goat grazes is symmetrical; this will tell you how long the segments AD , AB , and BC are, and thus what sort of triangle ABD must be. That will tell you how big the angle DAB is, and so how big angle DAC is. What fraction of the area of the circle centered on A is in the sector covering D to A (its center) to C and through B , and so what must that area be?

We aren't quite done, since we still have the slivers in the dark orange region where the goat can reach, but which aren't in the circular sector described above? What is the area of the triangle DAB? What is the area of the circular sector DAB? The difference is a sliver with the same size and shape as the two we're missing. The total area the goat can graze in is therefore the area of the big circular sector of the first paragraph plus two of those slivers.

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