The above figure shows a regular hexagram inscribed in a circle of radius 10 units. Find the area of the shaded region
The area of the circle is pi·radius2 = 100 pi
If you were to connect the center of the circle to each of the vertices of the interior hexagon, you would
get 6 congruent equilateral triangle and each of these triangles would be congruent to the six
equilateral triangles "at the points".
A radius to one of the points on the circle = 10, so the height of each of the equilateral triangles is 5.
This means that the base of the triangle is ( 5 / sqrt(3) ) · 2 = 10/sqrt(3)
The area of each triangle is ½ · ( 10/sqrt(3) ) · 5 = 25 / sqrt(3).
Since there are 12 congruent triangles, the total area is 300/sqrt(3) = 100·sqrt(3)
The area inside the circle but outside the hexagram = 100 pi - 100·sqrt(3)
Of this area, one-third is shaded = ( 100 pi - 100 sqrt(3) ) / 3