Challenge Level

A right-angled isosceles triangle whose two equal sides are 2 units in length is attached at its right-angled vertex to the centre of a square of side 2 units and rotated about this centre point.

What can you say about the area of the part of the square covered by the triangle as it rotates?

What happens to this area if the triangle is reduced in size so that its two equal sides are $ \sqrt 2 $ units?

What happens if the triangle is further reduced in size?