A circles area is pi times the square of its radius or area A of the circle is given by:
A = pi*r^2
So,if the radii of the two circles are a and b, then their areas are pi*a^2 and pi*b^2, where pi is a constant = circumference/(2*radius). So, the circles' areas with radius a and b, are in the ratio, pi*a^2 : pi*b^2 = a^2 : b^2.
Here, a = 4cm and b=7^cm. Therefore, the circles' areas are in the ratio, (4cm)^2 : (7cm)^2 or in the ratio 16:49
The area of a circle is directly given by the formula pi*(radius)^2. Thus the area of a circle is directly proportional to square of its radius.
Therefore ratio of areas of two circles with radii r1 and r2 will be equal to ratio of (r1)^2 and (r2)^2.
In the question it is given:
r1 = 4 cm and r2 = 7 cm.
Therefore ration of areas of these two circles is:
[(r1)^2]/[(r2)^2] = (4^2)/(7^2) = 16/49 = 0.32653