SOLUTION: What is the diameter of a circle circumscribed around a pentagon that is 8 in. on each side?

We draw in 2 lower adjacent radii, of length r. The central angle between 
two adjacent radii subtending a side of the pentagon are  or 72°
each:


Next we draw in a perpendicular to the bottom side,
(called an apothem) which divides the 8 inch bottom side into two
4 inch segments, and also divides the 72° central angle into two
36° angles:  



Now looking at the right triangle on the left, we have

sin(36°) =  = 

Multiply both sides by r

r·sin(36°) = 4

Divide both sides by sin(36°)

r = 

r =  

r = 6.905206467 inches

and the diameter is twice the radius, so

diameter of the circle = 2(6.905206467) = 13.61041293 inches

Edwin