The area of the circle is
We need to find the area of the triangle, subtract that area from the area of the circle and then divide it by 3.
Find the area of the equilateral triangle:
Draw the radius from O to B.
Draw the perpendicular bisector of BC through O intersecting BC at X.
Solve the right triangle we just made.
Hypotenuse = 4
Since it's a 30-60-90 right triangle, angle OBX = 30 degrees and angle BOX = 60 degrees.
With a little trig or 30-60-90 rules, we determine that OX = 2, and BX = .
Since triangle OBC is isosceles, BX = CX.
The base of the equilateral triangle ABC is .
All we need now is the height of the equilateral triangle.
We just add the radius 4 to OX and get 6.
The area of the inscribed equilateral triangle is .
Now you have both pieces I talked about in the beginning.
You may finish up now.
Turn the lights out when you leave.