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.