If x is the length of the side of the regular hexagon then 6x=30 which implies that the length of the sides of the hexagon is equal to 5. In order to find the radius of the circle that is inscribed in the hexagon we need to calculate the height h of (any) one of the equilateral triangles that are inside the hexagon and all of them have sides equal to 5. This is easily done by the use of the Pythagorean theorem, since the hypotenuse has length 5, one of the perpendicular sides has length 5/2 and the last perpendicular side has length h. Hence, h satisfies the equation h^2+(5/2)^2=5^2 or equivalently, h^2=25-25/4=(3)(25/4). Since h is the length of a side of a triangle it is positive and we therefore, obtain that h=(sqrt(3))(5/2) as the answer.
