Question 113684: Find the difference in the area of a circle circumscribed about a regular hexagon with sides of 10 cm and the area of the circle inscribed inside the same hexagon.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! You have a figure that consists of a hexagon, or 6-sided regular polygon, and two circles. One of the circles is circumscribed on the hexagon which means that the circle will intersect each of the vertices of the hexagon.
The interesting thing about a hexagon is that if you draw three straight lines connecting the opposite vertices, you will form 6 equilateral triangles inside of the hexagon. Since these triangles are equilateral, we now know that the radius of the circle is equal to the length of one of the hexagon's sides. So we can write:
 for the area of the circumscribed circle.
The inscribed circle is going to be tangent to the sides of the hexagon at the midpoint of the sides. That means that if you construct a radius for the inscribed circle, you will form a 30-60-90 right triangle where the hypotenuse is the radius of the circumscribed circle, the short leg is one-half of the side of the hexagon, and the long leg is the radius of the inscribed circle.
We know that the sides of a 30-60-90 right triangle are in proportion  : : , therefore the radius of the inscribed circle is:
 , so now we can write:
Now we need to compute the difference between the two areas, 
Hope that helps,
John
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