Question

Three congruent circles touch one another as shown in the figure. The radius of each circleis 6 cm. Find the area of the unshaded region within the triangle ABC

Answer 1

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

`\text{Area of triangle ABC = }\frac{\sqrt[]{3}}{4}a^2`

where a represents each side of the triangle.

In triangle ABC , a = 6 + 6

a = 12 cm

`\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}`

There are three sectors contained in the triangle, and each of them form an angle 60° with the center.

The radius, r = 6 cm

`\begin{gathered} \text{Area of each sector = }\frac{\theta{}}{360}*\pi r^2 \\ \text{Area of each sector = }(60)/(360)*\pi*6^2 \\ \text{Area of each sector = 6}\pi \end{gathered}`

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)