1. **State the problem:** We are given that the perimeter (circumference) of a circle is equal to the perimeter of a square. We need to find the ratio of their areas. 2. **Formulas used:** - Perimeter of a square: $P_{square} = 4s$, where $s$ is the side length. - Circumference of a circle: $P_{circle} = 2\pi r$, where $r$ is the radius. - Area of a square: $A_{square} = s^2$. - Area of a circle: $A_{circle} = \pi r^2$. 3. **Set the perimeters equal:** $$4s = 2\pi r$$ 4. **Express $s$ in terms of $r$:** $$s = \frac{2\pi r}{4} = \frac{\pi r}{2}$$ 5. **Calculate the ratio of areas:** $$\text{Ratio} = \frac{A_{circle}}{A_{square}} = \frac{\pi r^2}{s^2} = \frac{\pi r^2}{\left(\frac{\pi r}{2}\right)^2}$$ 6. **Simplify the denominator:** $$\left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4}$$ 7. **Substitute back:** $$\text{Ratio} = \frac{\pi r^2}{\frac{\pi^2 r^2}{4}} = \pi r^2 \times \frac{4}{\pi^2 r^2}$$ 8. **Cancel common terms:** $$= \frac{4}{\pi}$$ **Final answer:** The ratio of the areas of the circle to the square is $$\boxed{\frac{4}{\pi}}$$