A regular hexagon is inscribed in a circle as shown in the attached image.
Let side of regular hexagon be 'x'. So, each side of hexagon is 'x'.
Now, Perimeter of hexagon = 90 (GIVEN)
Perimeter of hexagon = Sum of all its sides
90=x+x+x+x+x+x
90=6x
So, x=15 units. Therefore, each side of regular hexagon is 15 units.
Join each vertex of the hexagon to its opposite vertex. The point of intersection of these lines is center of the circle say O.
Since, sum of all angles in a regular hexagon is 720 degrees. In regular hexagon each angle is equal. Let each angle be 'y'
y+y+y+y+y+y=[tex] 720^{\circ} [/tex]
6y=[tex] 720^{\circ} [/tex]
y=120 degrees
Since, [tex] \angle FAB = 120^{\circ} [/tex]
[tex] \angle FAO = 60^{\circ} [/tex]
Similarly [tex] \angle AFO = 60^{\circ} [/tex]
Therefore, [tex] \angle AOF = 60^{\circ} [/tex] (by angle sum prop)
Here triangle AOF is an equilateral triangle.
So, each side of triangle AOF is equal which is 15 units.
So, the radius is 15 units.
