The circumference of a circle is equal to the side of a square whose area measures 407044 sq. cm. What is the area of the circle? (Bank P.O., 2010)

### Concept & Formula The side of the square can be found from its area, which is then equal to the circumference of the circle. We can then find the radius and subsequently the area of the circle. $$ \text{Area of square} = a^2 $$ $$ \text{Circumference of circle} = 2\pi r $$ $$ \text{Area of circle} = \pi r^2 $$ ### Step-by-Step Solution 1. **Given:** Area of the square = $407044 \text{ sq. cm}$. 2. **Find the side of the square ($a$):** $a^2 = 407044 \Rightarrow a = \sqrt{407044} = 638 \text{ cm}$. 3. **Find the radius of the circle ($r$):** Circumference = Side of square $2\pi r = 638 \Rightarrow 2 \times \frac{22}{7} \times r = 638$ $r = \frac{638 \times 7}{44} = 101.5 \text{ cm}$. 4. **Find the area of the circle:** Area = $\pi r^2 = \frac{22}{7} \times (101.5)^2 = \frac{22}{7} \times 10302.25 = 22 \times 1471.75 = 32378.5 \text{ sq. cm}$. ### Exam Strategy & Shortcut When you reach the step of calculating $r$, you can estimate. $638 / (2 \pi) \approx 638 / 6.28 \approx 100$. The area is $\pi (100)^2 \approx 31415$. The closest option is 32378.5. ### Common Pitfall Miscalculating the square root of 407044 or the squaring of 101.5 are common errors. Always double-check your arithmetic when dealing with large numbers. ### Final Answer Therefore, the correct answer is **32378.5 sq. cm**.