Our user asked us to create a calculator which should determine the "side length of the regular polygon (pentagon, hexagon) by diameter or radius of a circumscribed circle".
Actually, this is quite simple. We have to find the length of the base of the triangle, which is formed by the center of the polygon and two adjusted vertexes of the regular polygon. Since the two remaining sides of the triangle are the two radii, and the center's angle is 360 divided by the number of sides of the regular polygon, we can use the law of sines - two sides related to each other as sines of opposite angles. Our triangle is also isosceles, so finding the remained angles is a piece of cake. You can see the result below.
P.S. Now, I've been asked how to find the regular polygon's side length by a radius of an inscribed circle?
This is even simpler. Our triangle is formed by the center of the polygon, one of the vertexes, and the contact point between the circle and the polygon side. The point of contact has a right angle, so half of the polygon side is radius multiplied by the tangent of a sharp angle. The sharp angle is 360 divided by the number of sides and divided by 2. You can see the result below.