The circumferences of two circles are 132 metres and 176 metres respectively. What is the difference between the area of the larger circle and the smaller circle?

### Concept & Formula The area of a circle can be determined once its radius is found from the given circumference. $$C = 2\pi r$$ $$A = \pi r^2$$ ### Step-by-Step Solution * Given: Circumference of the smaller circle, $C_1 = 132$ m, and larger circle, $C_2 = 176$ m. * Calculate the radius of the smaller circle ($r_1$): $2 \times \frac{22}{7} \times r_1 = 132 \Rightarrow r_1 = \frac{132 \times 7}{44} = 21$ m. * Calculate the radius of the larger circle ($r_2$): $2 \times \frac{22}{7} \times r_2 = 176 \Rightarrow r_2 = \frac{176 \times 7}{44} = 28$ m. * Calculate the difference in areas: $\text{Difference} = \pi r_2^2 - \pi r_1^2 = \pi (r_2^2 - r_1^2)$ $= \frac{22}{7} \times (28^2 - 21^2)$ $= \frac{22}{7} \times (28 - 21)(28 + 21)$ $= \frac{22}{7} \times 7 \times 49 = 22 \times 49 = 1078$ sq. m. ### Exam Strategy & Shortcut Instead of calculating the two areas separately and then subtracting, use the algebraic identity $a^2 - b^2 = (a-b)(a+b)$ to factorize $\pi(r_2^2 - r_1^2)$. This significantly reduces calculation time and avoids squaring large numbers. ### Common Pitfall Calculating $\pi \times 28^2$ and $\pi \times 21^2$ individually can lead to tedious multiplication errors. Factoring out $\pi$ is essential. ### Final Answer Therefore, the correct answer is **1078 sq. m**.