If the perimeter of a square is equal to the circumference of a circle then the ratio of their areas is

Let the side of the square be a and the radius of the circle be r.
Now, Perimeter of circle = Circumference of the circle

$\Rightarrow 4 a=2 \pi r$

$\Rightarrow \frac{a}{r}=\frac{\pi}{2}$

Now,

$\frac{\text { Area of square }}{\text { Area of circle }}=\frac{a^{2}}{\pi r^{2}}$

$=\frac{1}{\pi} \times\left(\frac{a}{r}\right)^{2}$

$=\frac{1}{\pi} \times \frac{\pi^{2}}{4}$

$=\frac{\pi}{4}$