A piece of wire 132 cm long is bent successively to form three different closed shapes: (1) an equilateral triangle, (2) a square, and (3) a circle. For which of these shapes will the enclosed area be greatest?

Introduction:
This is a conceptual geometry question about maximizing area for a given perimeter. We are given a fixed wire length and asked which shape encloses the largest possible area when the wire is bent into different forms. Understanding isoperimetric properties and regular shapes is crucial here.

Given Data / Assumptions:

• Wire length (perimeter or circumference) = 132 cm.
• Shapes formed: equilateral triangle, square, circle.
• Perimeter or circumference of each shape equals 132 cm.
• We must decide which shape has the largest area.

Concept / Approach:
Among all plane figures with a given perimeter, the circle encloses the maximum area. Among polygons with a fixed perimeter, a regular polygon with more sides encloses more area than one with fewer sides. Therefore, we expect the circle to have the largest area. We can also confirm this by computing approximate areas of each shape using the common perimeter.

Step-by-Step Solution:
For the equilateral triangle: perimeter = 3a = 132 ⇒ a = 44 cm. Area triangle Aₜ = (√3 / 4) * a² = (√3 / 4) * 44² = (√3 / 4) * 1936 ≈ 484√3 ≈ 838.6 cm². For the square: perimeter = 4s = 132 ⇒ s = 33 cm. Area square Aₛ = s² = 33² = 1089 cm². For the circle: circumference = 2πr = 132 ⇒ r = 132 / (2π) = 66 / π. Using π ≈ 3.14, r ≈ 21.02 cm. Area circle A𝚌 = πr² ≈ 3.14 * (21.02)² ≈ 3.14 * 441.8 ≈ 1387.2 cm².

Verification / Alternative check:
We see numerically that A𝚌 ≈ 1387.2 cm², Aₛ = 1089 cm², and Aₜ ≈ 838.6 cm². Therefore, the circle clearly has the largest area. This matches the theoretical result that the circle is the most area efficient shape for a fixed perimeter.

Why Other Options Are Wrong:
The equilateral triangle and the square, although regular, each enclose less area than the circle for the same perimeter. The areas are not equal across shapes, so the option stating equality is also incorrect. Only the circle consistently maximizes area with fixed perimeter.

Common Pitfalls:
Some learners think more sides always mean maximum area and stop at the square, forgetting that the circle can be seen as a polygon with infinitely many sides. Others may not know or recall the general isoperimetric property. A quick area comparison confirms the theoretical expectation.

Final Answer:
The enclosed area is largest when the wire is bent into a circle.