How would I calculate the area of the shaded region of a circle with radius 6 and length of chord AB is 6.

-
Do you know how to find the area of an equilateral triangle and a circular sector? –  Guess who it is. Jul 17 '12 at 8:47
Area of sector = area of shaded region + area of equilateral triangle –  Rajeshwar Jul 17 '12 at 8:51

Let $O$ be center of given circle. From assumption you have $\bar{OA}=\bar{OB}=6$. So you have all information to find: (a) area of triangle $ABO$ (b) angle between $\bar{OA}$ and $\bar{OB}$. Next use formula for area of fragment of a circle between points $A, B$ and $O$ say - P. Finally subtract area of area of the triangle $ABO$ from P and get the result.