# Goat tethered in a circular pen

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?

• Did you try and draw a picture? – Pedro Tamaroff Jun 6 '13 at 22:24
• Yes I did. It didn't help much. – Brad Jun 6 '13 at 22:27

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 $ADBC$.