Largest circle inside a square of side 18 cm (repaired stem):\nA wire is used to draw the largest possible circle that fits inside a square of side 18 cm. What is the area (in cm^2) of this circle?

Introduction / Context:
(Recovery–First applied) The original stem mentioned “sphere” with area options in terms of π. A sphere’s surface area would not match these choices for side 18 cm. The natural, minimal-repair interpretation is the largest circle that can be inscribed in the square. We therefore correct the stem to “largest circle inside a square.”

Given Data / Assumptions:

• Square side s = 18 cm.
• Inscribed circle has diameter = s, hence radius r = s/2 = 9 cm.

Concept / Approach:
An inscribed circle in a square touches all sides; its diameter equals the square’s side. Area of the circle is A = πr^2.

Step-by-Step Solution:

Verification / Alternative check:
If one attempted a sphere-in-cube interpretation, the correct surface area would be 4πr^2 with r = 9 → 324π, which is not offered. The circle interpretation gives 81π, which is present and consistent with “inside a square.”

Why Other Options Are Wrong:
972π, 36π, 288π, 100π do not equal π * 9^2 for the inscribed circle in a side-18 square.

Common Pitfalls:
Confusing inscribed circle’s radius with the square’s diagonal/2; here diameter equals side, not the diagonal.

Final Answer:
81π