SOLUTION: A regular hexagon is inscribed in a circle of radius 15...Determine the area of the shaded region. The hexagon is shaded, but the circle isn't. Please help? Thank you!

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Question 823461: A regular hexagon is inscribed in a circle of radius 15...Determine the area of the shaded region. The hexagon is shaded, but the circle isn't. Please help? Thank you!
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The side length of the hexagon and two sides from the central angle form an isosceles triangle, and you want the area of SIX of these triangles. The circle is cut into sectors for each central angle to be 360/6=60 degrees.

Now consider each isosceles triangle is composed of TWO right triangles of the 30-60-90 type. You have TWELVE of these triangles making the hexagon. The hypotenuse is 15 units. The long leg is 2 times the short leg. If the short leg is x, and the long leg is 2x, then:

x%5E2%2B%282x%29%5E2=15%5E2
4x%5E2%2Bx%5E2=15%5E2
5x%5E2=15%2A15
x%5E2=45
x=3%2Asqrt%285%29

The area of ONE isosceles triangle is %281%2F2%29%28base%29%28height%29; which is
%281%2F2%29%282%2A3%2Asqrt%285%29%29%282%2A3%2Asqrt%285%29%29
%281%2F2%2936%2A5
18%2A5
90 square units.
'
Six of these is 6%2A90=270 square units.