Ratio of area of circle to square

Let that $l = 2\pi r$ . Then the area of the rolled-out circle is $A_\bigcirc = \pi r^2$ . The area of the rolled-out square is $A_\square = \left(\dfrac {2\pi r} 4\right)^2 = \dfrac {\pi^2 r^2}4$ . Then $\dfrac {A_\bigcirc}{A_\square} = \dfrac {\pi r^2}{\frac {\pi^2 r^2}4} = \dfrac 4\pi \approx \boxed{1.273}$ .

When two straight lines of length "l" is rolled into a circle and then a square the perimeters are the same.

Therefore 2 * pi * r = 4 * s or r/s = 2/pi where r is the radius of the circle and s is the length of each side of the square.

The ratio of the areas of the circle to the square is equal to pi * r * r/s * s = p i* (r/s) ^2 =pi *(2/pi) ^2 = 4/pi = 1.273