There are three circles each of radius $\sqrt{7}$ cm. A triangle is formed by joining their centres. The angles at the centre made by the triangle are shown in the figure. The area of the shaded portion is

### Concept & Formula The shaded portion consists of three separate sectors from three identical circles. When the centers of three circles are joined to form a triangle, the sum of the interior angles of the triangle is always $180^\circ$. Therefore, the sum of the central angles of the three shaded sectors is $180^\circ$. $$Total\ Area = \frac{\theta_1 + \theta_2 + \theta_3}{360^\circ} \times \pi r^2$$ ### Step-by-Step Solution * Given radius $r = \sqrt{7} \text{ cm}$. * The angles shown in the figure are $42^\circ$, $58^\circ$, and $80^\circ$. * Sum of the angles = $42^\circ + 58^\circ + 80^\circ = 180^\circ$. * The sum of the areas of the three sectors is equivalent to the area of a single sector with an angle of $180^\circ$ (which is a semi-circle). * Total Shaded Area $= \frac{180}{360} \times \pi \times r^2$ * Area $= \frac{1}{2} \times \frac{22}{7} \times (\sqrt{7})^2$ * Area $= \frac{1}{2} \times \frac{22}{7} \times 7$ * Area $= \frac{1}{2} \times 22 = 11 \text{ cm}^2$. ### Exam Strategy & Shortcut You do not need to calculate the area of each sector individually. Because the circles have identical radii and form a triangle, the sectors always sum up to exactly half a circle ($180^\circ$), regardless of the specific individual angles shown. Total area is just $\frac{1}{2} \pi r^2$. ### Common Pitfall Trying to calculate the three sector areas separately and then summing them up will take far too much time and increases the likelihood of arithmetic errors. ### Final Answer Therefore, the correct answer is **11 cm²**.