Answer:
Option A is correct
Area of the unshaded region = 18(2√3 - π)
Explanations:
Note that triangle ABC is an equilateral triangle, therefore the area of triangle ABC will be found using the formula for the area of an equilateral triangle
[tex]\text{Area of triangle ABC = }\frac{\sqrt[]{3}}{4}a^2[/tex]
where a represents each side of the triangle.
In triangle ABC , a = 6 + 6
a = 12 cm
[tex]\begin{gathered} \text{Area of triangle ABC = }\frac{\sqrt[]{3}}{4}(12^2) \\ \text{Area of triangle ABC = }\frac{144\sqrt[]{3}}{4} \\ \text{Area of triangle ABC = }36\sqrt[]{3} \end{gathered}[/tex]
There are three sectors contained in the triangle, and each of them form an angle 60° with the center.
The radius, r = 6 cm
[tex]\begin{gathered} \text{Area of each sector = }\frac{\theta{}}{360}\times\pi r^2 \\ \text{Area of each sector = }\frac{60}{360}\times\pi\times6^2 \\ \text{Area of each sector = 6}\pi \end{gathered}[/tex]
Area of the three sectors contained in the triangle = 3(6π)
Area of the three sectors contained in the triangle = 18π
Area of the unshaded region = (Area of the triangle ABC) - (Total Area of the sectors)
Area of the unshaded region = 36√3 - 18π
Area of the unshaded region = 18(2√3 - π)
