Perimeter preserved — A square has area 64 sq cm. A circle with the same perimeter (circumference) is formed. Find its area.

Introduction / Context:
We map a fixed perimeter from a square to a circle. The square perimeter becomes the circle circumference. From the circumference we get the circle radius and hence its area.

Given Data / Assumptions:

• Square area = 64 ⇒ side s = 8 cm
• Square perimeter = 4s = 32 cm
• Circle circumference = 32 = 2πr

Concept / Approach:
From 2πr = 32, get r = 16/π. Then circle area A = πr^2 = π * (16/π)^2 = 256/π sq cm.

Step-by-Step Solution:
r = 16/πA = π * (256/π^2) = 256/π sq cm

Verification / Alternative check:
Units consistent; the transformation uses perimeter equality only, not area equality.

Why Other Options Are Wrong:
Other fractions do not follow from r = 16/π. The exact symbolic form 256/π is required.

Common Pitfalls:
Using square area as circle area or using diameter instead of circumference to compute r.

Final Answer:
256/π sq cm