How to find an area of a circumbscribed hexagon using the diameter?

The formula for the area of a circumscribed hexagon  = 1/2 * 6 * a * r    where a = side length and r = radius of the circle  ( = 1/2 the diameter)  so in terms of the diameter and side length its 1/2 * 1/2 * 6 *d * a  =   1.5 ad.

You can calculate the value of a  by using trigonometry  You consider one of the 6 isosceles triangles that make up the hexagon. The vertex angle of these triangles = 60 degrees so:-

tan 30  =  1/2 a /  r  =  1/2 a / 1/2 d  = a/d

a =  d tan 30  = sqrt3 d / 3

Substituting for a :- 

So the  required formula for the area of the hexagon = 1.5 ad 

= 1.5 * sqrt3 d * d / 3  =  1/2 sqrt3 d^2   Answer