If the circumference of a circle is increased by 20%, what will be the effect on the circle?

### Concept & Circumference and Area Proportionality An increase in the circumference by a certain percentage implies the exact same percentage increase in the radius, since $C = 2\pi r$. The effect on the area (which is implied by "effect on the circle" given the options) can be calculated using the successive percentage change formula: $$x + y + \frac{xy}{100}$$ ### Step-by-Step Solution * **Given:** Circumference increases by $20\%$. * **Deduction:** Because $C \propto r$, the radius also increases by $20\%$. So, $x = 20$. * **Calculation:** We need to find the change in area ($A = \pi r^2$). Apply successive change for two identical $20\%$ increases: $20 + 20 + \frac{20 \times 20}{100}$. * **Calculation:** $40 + \frac{400}{100}$. * **Calculation:** $40 + 4 = 44\%$. ### Exam Strategy & Shortcut Any $20\%$ linear increase always leads to a $44\%$ area increase. Memorize standard values: $10\% \rightarrow 21\%$, $20\% \rightarrow 44\%$, $30\% \rightarrow 69\%$. This saves calculation time entirely. ### Common Pitfall Students often select "Cannot be determined" because the question vaguely asks for "effect on the circle" rather than specifically "effect on the area". However, analyzing the numerical options makes it clear area is being requested. ### Final Answer Therefore, the correct answer is **44% increase**.