Age 16 to 18 Challenge Level:
In the equilateral triangle, draw in the altitudes of the triangle
and taking the radius of the circle as $1$ unit, calculate the
In the square it is easiest to take the side of the square as $1$
unit and then calculate the radius of the circle.
In the pentagon it is easiest to take the side of the pentagon as
$1$ unit, and the chord length $XZ$ as $x$ units and then use
properties of similar triangles. Derive a quadratic equation which
you can solve to find $x$.