Three circles of radius $3.5\text{ cm}$ are placed in such a way that each circle touches the other two. The area of the portion enclosed by the circles is

### Concept & Area Enclosed by Touching Circles When three equal circles touch each other, connecting their centers forms an equilateral triangle. The area enclosed between the circles is the area of this triangle minus the area of the three circular sectors within it. $$\text{Enclosed Area} = \text{Area of Equilateral Triangle} - \text{Area of 3 Sectors}$$ ### Step-by-Step Solution * **Given:** Radius $r = 3.5\text{ cm} = \frac{7}{2}\text{ cm}$. * **Deduction:** The side of the equilateral triangle formed by the centers is $a = 2r = 7\text{ cm}$. * **Calculation:** Area of the equilateral triangle = $\frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} \times 7^2 = \frac{49\sqrt{3}}{4}$. * **Calculation:** Using $\sqrt{3} \approx 1.732$, Area $= \frac{49 \times 1.732}{4} \approx 21.217\text{ cm}^2$. * **Calculation:** Each interior angle is $60^\circ$. The three sectors form a semicircle ($180^\circ$). Area of sectors = $\frac{1}{2}\pi r^2 = \frac{1}{2} \times \frac{22}{7} \times (\frac{7}{2})^2 = \frac{11}{7} \times \frac{49}{4} = \frac{77}{4} = 19.25\text{ cm}^2$. * **Calculation:** Enclosed Area $= 21.217 - 19.25 = 1.967\text{ cm}^2$. ### Exam Strategy & Shortcut For three mutually touching circles of radius $r$, the standard formula for the enclosed area is $r^2(\sqrt{3} - \frac{\pi}{2})$. Plugging in $r = 3.5$ directly yields $(3.5)^2(1.732 - 1.571) = 12.25 \times 0.161 = 1.972$. Due to approximation differences in $\pi$ and $\sqrt{3}$, $1.967$ is the intended precise answer using $\frac{22}{7}$. ### Common Pitfall A common mistake is failing to convert the three $60^\circ$ sectors into a single semi-circle for calculation, leading to overly complex and error-prone fraction math. ### Final Answer Therefore, the correct answer is **$1.967\text{ cm}^2$**.