Two goats are tethered to adjacent corners of a 10-by-10 square...

Question

Two goats are inside a 10-by-10 square pasture. They are each on a leash of length 10, tied to poles at neighboring corners. What is the area of the pasture that neither goat can reach?

I almost solved it by labeling areas of the circles with variables, and then writing equations for the area of the circles and the square, and then subtracting, but that doesn't really get me anywhere.

Answer 1

The part of the pasture that at least one goat can reach can be broken into

• an equilateral triangle of side 10, which has an area of $\frac{\sqrt3}4\cdot10^2=25\sqrt3$
• two circular sectors of radius 10 and combined angle $\frac\pi3$, which have an area of $\frac{\pi\cdot10^2}6=\frac{50\pi}3$

Hence the area of the part of the pasture neither goat can reach is $$100-25\sqrt3-\frac{50\pi}3=4.3388522\dots$$