Your circle with regular pentagonal star.
1) Construct Angle Bisector at A
2) Construct Angle Bisector at B
3) Notice that you also just constructed the perpendicular bisector of AC.
4) Angle bisector from B intersects AC at point F. Label it.
5) The angle bisectors intersect at the Center of the circle. Call this O.
6) You have a right triangle, CFO.
7) Hypotenuse is Circle Radius.
8) Interior Acute Star angles are 36º. Half of that is 18º
9) cos(18º) = 5 / Radius.
I have no doubt there is an easier way.