If the ratio between the areas of two circles is 4 : 1 then the ratio between their radii will be

### Concept & Formula The area of a circle is proportional to the square of its radius. Therefore, the ratio of the areas of two circles is equal to the square of the ratio of their radii. $$ \frac{\text{Area}_1}{\text{Area}_2} = \frac{\pi r_1^2}{\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2 $$ ### Step-by-Step Solution 1. **Given:** The ratio of the areas of two circles is $4 : 1$. 2. **Setup the equation:** Let the radii be $r_1$ and $r_2$. $\frac{\pi r_1^2}{\pi r_2^2} = \frac{4}{1}$ 3. **Solve for the ratio of radii:** $\left(\frac{r_1}{r_2}\right)^2 = \frac{4}{1}$ $\frac{r_1}{r_2} = \sqrt{\frac{4}{1}} = \frac{2}{1}$. So, the ratio is $2 : 1$. ### Exam Strategy & Shortcut Whenever given the ratio of areas of two 2D similar shapes (like circles or squares), simply take the square root of the ratio to find the ratio of their corresponding linear dimensions (radii, diameters, circumferences). $\sqrt{4} : \sqrt{1} = 2 : 1$. ### Common Pitfall A common mistake is forgetting to take the square root or accidentally inverting the ratio (e.g., answering $1 : 2$ instead of $2 : 1$). Pay attention to the order in the problem statement. ### Final Answer Therefore, the correct answer is **2 : 1**.