Find the area of the region between a regular hexagon with sides of 6" and its inscribed circle.
Accepted Solution
A:
You can compute the area of the region between the hexagon and the circle by subtracting:
Area of region = Area of hexagon - Area of inscribed circle
Let's find those areas.
If we draw a segment from the center of the hexagon to each of its vertex, the hexagon will be divided into 6 equal triangles. For any of those triangles, the angle in the center equals 360°/6 = 60°. Being the other angles equal, all three angles must equal 60° (because they must add up to 180°). Hence, those triangles are equilateral. The height (h) from the center divides the triangle into two equal right triangles. We know the hypotenuse is 6, because of the equilateral triangle we began with, and one of the other sides is half the side of the hexagon, 3. Applying the Pythagorean Theorem to this right triangle, we find that its height is: