Consider a standard chess board. What is the diameter of the largest circle that can be drawn on the board whilst only drawing on the black squares.

The circle could fit entirely with in one black square, it could be like the green circle below, but we want the biggest, and that is the red circle:-




There is some logic as to why this is the largest circle that can be drawn. It has to do with the fact that a circle can only intercept a straight line at two points, you can visualise a square around our red circle travelling through the corners in the same place. (you could calculate the dimensions of this too.)

Using the yellow triangle we can easily calculate the radius and therefore the diameter of the circle. We need the hypotenuse of a triangle with opposite and adjacent sides of 1.5 and 0.5

Radius^2 = 1.5^2 + 0.5^2

Radius^2 = 10/4

Radius = Sqrt(10) / 2

Diameter = Sqrt(10) = 3.1623

Source : puzzles.nigelcoldwell.co.uk/thirtyonehint.htm