Circle on a square’s diagonal: A square has area 50 square units. A circle is drawn with the square’s diagonal as its diameter. Find the circle’s area.

Introduction / Context:
This mixes properties of squares and circles: from a square’s area, deduce its side and diagonal; that diagonal becomes the circle’s diameter. Then compute the circle’s area from its radius (half the diameter).

Given Data / Assumptions:

• Square area = 50
• Side s = √50
• Diagonal d = s * √2
• Circle diameter = d; radius r = d/2

Concept / Approach:
Compute s from area. Use the square’s diagonal relation d = s√2. Halve d for the circle’s radius and apply A_circle = πr^2. Keep results exact in terms of π to avoid rounding errors.

Step-by-Step Solution:
s = √50d = √50 * √2 = √100 = 10r = d/2 = 10/2 = 5A_circle = π * 5^2 = 25π

Verification / Alternative check:
Back-check: A_square = 50, so s ≈ 7.071. Diagonal ≈ 10, radius ≈ 5 → area ≈ 78.54, which equals 25π numerically.

Why Other Options Are Wrong:
50π and 100π double-count; 12.5π halves incorrectly; “None” is false because 25π is exact.

Common Pitfalls:
Using side as diameter; forgetting the √2 factor for a square’s diagonal; squaring before or after halving incorrectly.

Final Answer:
25π sq. units