If the radius of a circle is doubled, its area is increased by

### Concept & Scaling Factors in Geometry When the linear dimension (radius) of a 2D shape is scaled by a factor of $k$, its area is scaled by a factor of $k^2$. $$\text{New Area} = k^2 \times \text{Old Area}$$ ### Step-by-Step Solution * **Given:** The radius is doubled. This means the scaling factor $k = 2$. * **Calculation:** Let the original radius be $r$. Original area $A_1 = \pi r^2$. * **Calculation:** New radius is $2r$. New area $A_2 = \pi (2r)^2 = 4\pi r^2$. * **Deduction:** The new area is $4$ times the original area. * **Calculation:** The *increase* in area is $4\pi r^2 - \pi r^2 = 3\pi r^2$. * **Calculation:** Percentage increase = $(\frac{3\pi r^2}{\pi r^2}) \times 100 = 300\%$. ### Exam Strategy & Shortcut "Doubled" means a multiplier of $2$. Area multiplier is $2^2 = 4$. If you have $4$ times the amount you started with, you have gained $3$ times your original amount. $3 \times 100\% = 300\%$ increase. ### Common Pitfall A classic trap is selecting $400\%$. The area *becomes* $400\%$ of its original size, but the question specifically asks what it is *increased by*, which requires subtracting the original $100\%$. ### Final Answer Therefore, the correct answer is **300%**.