The area of a square and the area of a rectangle are equal. If the side of the square is 40 cm and the length of the rectangle is 64 cm, find the perimeter of the rectangle in cm, assuming both shapes have exactly the same area.

Introduction / Context:
This problem tests the relationship between area and perimeter across different shapes. You are told that a square and a rectangle have the same area. The square’s side gives you its area directly. Then, using the rectangle’s length, you can compute its breadth from area = length * breadth. Finally, apply the rectangle perimeter formula 2 * (length + breadth).

Given Data / Assumptions:

• Side of square = 40 cm
• Area of square = side * side
• Rectangle length = 64 cm
• Areas are equal, so rectangle area = square area
• Perimeter of rectangle = 2 * (length + breadth)

Concept / Approach:
Compute square area. Use it as rectangle area. Find breadth = area / length. Then compute perimeter from length and breadth.

Step-by-Step Solution:

Verification / Alternative check:
Check area consistency: 64 * 25 = 1600 sq cm, which matches the square’s area. Since the area match is correct, the perimeter calculation using those dimensions is valid.

Why Other Options Are Wrong:

Common Pitfalls:
Common mistakes include using perimeter formulas for area, mixing up square side with rectangle length, or calculating breadth as 64/1600 instead of 1600/64. Also, always keep units consistent in cm throughout the calculation.

Final Answer:
178 cm