A pig is in a field that is an equilateral triangle with sides 100 metres long. The pig is tethered by a rope to one corner of the field and it can graze exactly half the area of the field.

How long is the rope?

We can simplify this puzzle by making six triangular fields and forming these to make a hexagon. Then we are trying to find the radius of a circle that is half the area of the hexagon.

We know that the sides of the hexagon are 100 metres long, so the area of the hexagon is

$$ 10000 x 3 \sqrt{3} / 2 $$

and half of that is

$$ 7500 \sqrt{3} $$

The area of the circle is given by π r squared. So the radius r is

$$ \sqrt{(7500 \sqrt{3} / \pi)} $$

Which is approximately 64.3 metres.