This video shows how to construct a square inside a circle, using only a ruler and a compass. The circle become the circumcircle of the square. This constructed was selected by a computer to have as few steps as possible. The square ends up at an attractively chunky angle relative to the original radius of the circle.
For more videos and math help, see http://www.dr-mikes-math-games-for-kids.com/
Music "Half Bit" by Kevin MacLeod, http://www.incompetech.com