If the circumference of a circle increases form $4\pi$ to $8\pi$, what change occurs in its area?

### Concept & Circumference and Area Relationship The scale factor for the circumference of a circle is identical to the scale factor for its radius. Once the scale factor for the radius ($k$) is known, the area scales by a factor of $k^2$. $$k = \frac{C_2}{C_1} = \frac{r_2}{r_1}$$ ### Step-by-Step Solution * **Given:** Initial circumference $C_1 = 4\pi$. Final circumference $C_2 = 8\pi$. * **Calculation:** Determine the scale factor for circumference: $k = \frac{8\pi}{4\pi} = 2$. * **Deduction:** Since circumference doubles, the radius also doubles. * **Calculation:** Let original area be $A_1 = \pi r^2$. * **Calculation:** If radius doubles, new area $A_2 = \pi(2r)^2 = 4\pi r^2$. * **Deduction:** The new area is $4$ times the original area, which means it quadruples. ### Exam Strategy & Shortcut Skip finding the actual radius. Notice that $8\pi$ is twice $4\pi$. The linear dimensions (circumference, radius, diameter) doubled ($2\times$). Area is 2-dimensional, so it scales by $2^2 = 4\times$. Thus, it quadruples. ### Common Pitfall A common mistake is assuming that area increases linearly with circumference and choosing "It doubles." Remember area is proportional to the square of linear dimensions. ### Final Answer Therefore, the correct answer is **It quadruples.**