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
lengths.

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 $x$.

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 $x$.