First notice that the drawn triangle is equilateral, as it has two congruent sides — the radii of a circle, with an included angle of 60^(∘). This means that the height of this triangle, 2sqrt(3), divides it into two 30^(∘)-60^(∘)-90^(∘) triangles.
Let's recall that in a 30^(∘)-60^(∘)-90^(∘) triangle the length of the longer leg is sqrt(3) times the length of the shorter leg. Since in our exercise the longer leg is 2 sqrt(3), the shorter leg of this triangle is 2.
Next recall that the length of the hypotenuse of the 30^(∘)-60^(∘)-90^(∘) triangle is 2 times the length of the shorter leg. Therefore the hypotenuse, which is also a radius of a circle, is 4.
Now we have all information needed to find the area of the triangle and the area of the sector of the circle.
A_s=60^(∘)/360^(∘)*π (4)^2≈ 8.38
A_t=1/2(4)( 2 sqrt(3))≈ 6.93
Finally, we will subtract the area of the triangle from the area of the circle.
As-A_t=8.38-6.93= 1.45
The area of the shaded region is approximately 1.45 square units.
