The shaded area can be determined by subtracting the area of the circle from the area of the hexagon. To accomplish this we will first draw diagonals between opposite vertices, thereby creating 6 congruent isosceles triangles. In one of these triangles we will also draw its height, h.
By determining the area of one triangle we can find the area of the hexagon if we multiply this number by 6. Also, notice that the height of the triangle is the circle's radius. Therefore, finding the height also gives us the circle's radius.
Finding the Height
A hexagon has 6 sides. By substituting the number of sides in the formula 180^(∘)(n-2)n, where n is the number of sides, we can find the measure of the hexagon's interior angles.
180^(∘)( 6-2)/6=120^(∘)
Since the diagonals bisect the interior angles, the height is also the longer leg in a 30^(∘)-60^(∘)-90^(∘) triangle. In this kind of special triangle the long leg is sqrt(3) times longer then the short leg.
Finding the Shaded Region
When we know the height of the triangle and circle's radius we can calculate the area of the hexagon and circle.
Hexagon:& (1/2* 3sqrt(3)* 6)6= 54 sqrt(3) in^2 [0.7em]
Circle:& π(3sqrt(3))^2 = 27π in^2
Finally, we calculate the shaded region's area by subtracting the circle's area from the hexagon's area.
Shaded region: 54 sqrt(3)-27π ≈ 8.71 in^2
