Let the square we are considering have a side of length L.
The incircle of a square touches all the four sides. The diameter of the circle is equal to the length of the square's side.The radius of the circle is L/2.
The circumcircle of a square touches all the four vertices of the circle. The diameter of the circle is equal to the length of the diagonal of the square.
Using the Pythagorean Theorem, the length of the diagonal of a square in terms of the length of its sides is sqrt(2*L^2) = (sqrt 2)*L. The radius of the circumcircle is L/(sqrt 2)
The area of a circle with radius r is pi*r^2.
The ratio of the area of the incircle to that of the area of the circumcircle of the square is [pi*(L/2)^2]/[pi*L^2/(sqrt 2)^2]
The correct answer is option C. The ratio of the are of the incircle of a square and that of the area of the circumcircle is 1:2