Areas of two circles are equal. Is it necessary that their circumferences are equal? Why?

(A) Yes,it is necessary that their circumferences are equal.
Let the radii of the two circles be $r_1$ and $r_2$ respectively.
The area of a circle is given by the formula $A = \pi r^2$.
Given that the areas are equal,we have $\pi r_1^2 = \pi r_2^2$.
Dividing both sides by $\pi$,we get $r_1^2 = r_2^2$,which implies $r_1 = r_2$ (since radius must be positive).
The circumference of a circle is given by $C = 2\pi r$.
Since $r_1 = r_2$,it follows that $2\pi r_1 = 2\pi r_2$.
Therefore,the circumferences of the two circles must be equal.