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

True

We are given that

Area of circle with radius R₁ = Area of circle with radius R₂

⇒ π(R₁)² = π(R₂)²

⇒ R₁= R₂

⇒ 2πR₁= 2πR₂, and hence the circumferences are also equal.