A square, a circle, and an equilateral triangle have the same perimeter. Consider statements: I. The square’s area is greater than the triangle’s area. II. The circle’s area is less than the triangle’s area. Which statement(s) is/are correct?

Introduction / Context:
Among plane figures with a fixed perimeter, the circle encloses the maximum possible area (isoperimetric inequality). For regular polygons with the same perimeter, more sides mean more area, approaching the circle’s maximum. Therefore, with the same perimeter, the circle has the largest area, followed by the square, then the equilateral triangle.

Given Data / Assumptions:

• Same perimeter for all three shapes: circle, square, equilateral triangle.
• Standard area-perimeter relationships for regular shapes.

Concept / Approach:
Use the isoperimetric principle qualitatively: Area(circle) > Area(square) > Area(equilateral triangle) for a fixed perimeter. This ordering is sufficient to evaluate the two statements.

Step-by-Step Reasoning:
Statement I: Square’s area > triangle’s area ⇒ True by the ordering.Statement II: Circle’s area < triangle’s area ⇒ False; in fact, circle’s area is the largest.

Verification / Alternative check:
Concrete example with a chosen perimeter confirms the inequality chain numerically (omitted for brevity since the inequality is standard).

Why Other Options Are Wrong:
“Only II” and “Both” contradict the isoperimetric ranking; “Neither” is incorrect because statement I is true.

Common Pitfalls:
Assuming equal side lengths rather than equal perimeters; confusing the role of “regularity” and the fact that among all plane curves of fixed perimeter, the circle is maximal in area.

Final Answer:
Only I