Between a square of perimeter 44 cm and a circle of circumference 44 cm, which figure has larger area and by how much?

### Concept & Isoperimetric Inequality For a given fixed perimeter, a circle will always enclose the maximum possible area compared to any other standard polygon (like a square or triangle). $$\text{Area}_{\text{square}} = s^2$$ $$\text{Area}_{\text{circle}} = \pi r^2$$ ### Step-by-Step Solution * **Analyze the Square:** Perimeter = $4s = 44$ cm $\Rightarrow s = 11$ cm. Area of Square = $s^2 = 11^2 = 121$ cm². * **Analyze the Circle:** Circumference = $2\pi r = 44$ cm. $2 \times \frac{22}{7} \times r = 44 \Rightarrow \frac{44}{7} \times r = 44 \Rightarrow r = 7$ cm. Area of Circle = $\pi r^2 = \frac{22}{7} \times 7^2 = 22 \times 7 = 154$ cm². * **Compare the Areas:** The circle has the larger area: $154$ cm² vs $121$ cm². Difference = $154 - 121 = 33$ cm². ### Exam Strategy & Shortcut Memorize standard integer pairs for speed: A circle with radius $r=7$ has circumference $C=44$ and Area $A=154$. A square with perimeter $P=44$ has side $s=11$ and Area $A=121$. Subtracting them instantly yields 33. ### Common Pitfall Assuming that figures with equal perimeters must logically hold equal areas, prompting a rapid but incorrect selection of option (a). ### Final Answer Therefore, the correct answer is **Circle, 33 cm²**.