A circle and a square have the same area. The ratio of the side of the square and the radius of the circle is

### Concept & Logic Equating the area formulas for a circle and a square allows us to find the direct proportional relationship between the side of the square ($a$) and the radius of the circle ($r$). $$ \pi r^2 = a^2 $$ ### Step-by-Step Solution 1. **Given:** Area of circle = Area of square. $\pi r^2 = a^2$ 2. **Find the ratio of side ($a$) to radius ($r$):** Divide both sides by $r^2$: $\frac{a^2}{r^2} = \pi$ 3. **Take the square root of both sides:** $\frac{a}{r} = \sqrt{\pi}$ $\frac{a}{r} = \frac{\sqrt{\pi}}{1}$ So, the ratio of the side of the square to the radius of the circle is $\sqrt{\pi} : 1$. ### Exam Strategy & Shortcut Set area to a simple constant like $\pi$. If area = $\pi$, then $a^2 = \pi \Rightarrow a = \sqrt{\pi}$. For the circle, $\pi r^2 = \pi \Rightarrow r = 1$. The ratio $a : r$ is immediately $\sqrt{\pi} : 1$. ### Common Pitfall Reversing the requested ratio is a common trap. The question asks for "side of the square" compared to the "radius of the circle", so ensure $a$ is the numerator/first term. ### Final Answer Therefore, the correct answer is **$\sqrt{\pi} : 1$**.