If we are going to take a look at the big picture:
(Area of the circle) - (Area of the hexagon) = Area of the shaded region.
Given:
A regular hexagon is composed of 8 congruent equilateral triangles, therefore:
r = 18
Area of the circle = π r2 = π (18)2 = 324 π
To get the area of the hexagon, first we have to get the height of one equilateral triangle, which is also called the apothem.
If apothem is drawn, you will see that it creates a 30-60-90 triangle with side lengths k- k√3 - 2k.
2k =18
k = 9
k√3 = 9√3 (This is the apothem.)
Area of the hexagon = (1/2) (apothem) (perimeter) = (1/2) (9√3) (18*6) = 486√3 ≈ 841.78
Area of the shaded region = 324 π - 486√3 ≈ 176.1 sq. unit.
Jon S.
Don't understand how a regular hexagon has 8 congruent equilateral triangles - I thought it was 6. Also how does that least to r=18?07/16/20