A piece of wire 132 cm long is bent successively to form three different closed plane figures: an equilateral triangle, a square, and a circle. In which of these three shapes will the enclosed area be the greatest when the entire wire is used each time?

Introduction / Context:
This question examines the relationship between perimeter and area for different plane figures. When we use the same length of wire to form different shapes, each shape will have the same perimeter but a different enclosed area. The goal is to know which shape maximizes the area for a given fixed perimeter. This concept is very important in optimization problems in geometry and appears frequently in aptitude tests.

Given Data / Assumptions:

• Length of the wire = 132 cm.
• The wire is bent into three shapes: an equilateral triangle, a square, and a circle.
• In each case, the entire 132 cm is used as the perimeter of that shape.
• We compare only the areas enclosed by these shapes, not their perimeters (which are the same).
• No material is lost or added when changing from one shape to another.

Concept / Approach:
A well-known geometric result states that among all closed plane figures with the same perimeter, the circle encloses the maximum possible area. Among polygons with a fixed perimeter, regular polygons (all sides equal and all angles equal) have larger areas than irregular ones, and as the number of sides increases, the area approaches that of the circle. Here, all three shapes are regular (equilateral triangle, square, circle), but the circle is known to give the largest area for the same perimeter, so we can answer the question using this theoretical fact without performing detailed numerical calculations.

Step-by-Step Solution:
Perimeter of each shape = 132 cm (by construction).For an equilateral triangle with perimeter 132 cm, each side is 132 / 3 = 44 cm.For a square with perimeter 132 cm, each side is 132 / 4 = 33 cm.For a circle with circumference 132 cm, radius r is found from 2 * π * r = 132.Although we could calculate the exact areas of each shape, we can rely on the theorem that for a given perimeter, the circle maximizes the enclosed area.Thus, the circle made from the wire will enclose more area than the equilateral triangle and the square formed from the same wire.

Verification / Alternative check:
If we wish, we can verify this numerically. For the square, area = side^2 = 33^2 = 1089 square centimetres. For the equilateral triangle with side 44 cm, area = (sqrt(3) / 4) * 44^2 ≈ 0.433 * 1936 ≈ 838.7 square centimetres, which is less than the area of the square. For the circle, using an approximate value of π, the radius r is about 132 / (2 * π) ≈ 21.0 cm. Then area of the circle is approximately π * r^2 ≈ 3.14 * 441 ≈ 1384 square centimetres, which is clearly greater than both 1089 and 838.7, confirming that the circle has the largest area.

Why Other Options Are Wrong:
The equilateral triangle and the square both enclose less area than the circle for the same perimeter, as shown by theory and numerical calculation. The option claiming equal areas in all shapes is incorrect because area depends on how the perimeter is distributed in the shape. The option stating that the result cannot be determined is also incorrect, since classical geometry directly answers this question and even numeric calculations confirm it.

Common Pitfalls:
Some learners mistakenly think that the shape with more sides always has more area, or that all shapes with the same perimeter have the same area. Others might incorrectly assume that a square and a circle give nearly the same area and might guess rather than reason. Remember that the circle is uniquely optimal for maximum area at a fixed perimeter, and this fact is widely used in design and optimization problems.

Final Answer:
The shape in which the wire encloses the greatest area is the circle.