A builder has 100 m of fencing wire. He can use this wire to enclose a circular garden or a square garden. If he wants to maximize the enclosed area, what is the approximate ratio of the area of the largest possible circular garden to the area of the largest possible square garden, using the entire 100 m of wire?

Correct — B. Use all 100 m as the perimeter of each shape in turn.

Circle radius: 2πr = 100 → r = 50⁄π ≈ 15.92 m
Circle area: πr² = 2500⁄π ≈ 795.8 m²

Square side: 100 ÷ 4 = 25 m
Square area: 25² = 625 m²

Divide: (2500⁄π) ÷ 625 = 4⁄π ≈ 1.27 : 1 → option (b)

Why the others are wrong

• (a)
  1:1 — Equal perimeters do not mean equal areas. The circle encloses about 795.8 m² against the square's 625 m², so the ratio is well above 1.
• (c)
  1.57:1 — 1.57 ≈ π⁄2 is the ratio of a circle to the square drawn inside it, where the diameter equals the square's diagonal. Here the shapes share a perimeter, which gives 4⁄π ≈ 1.27.
• (d)
  2:1 — A 2 : 1 ratio needs a 1250 m² circle. A 100 m circumference encloses only about 795.8 m², roughly 27% more than the square, not double.

For a fixed perimeter P, a square encloses P² ⁄ 16 and a circle encloses P² ⁄ 4π. Their ratio, 16 ⁄ 4π = 4⁄π ≈ 1.273, does not depend on the length of wire.

It reflects a general result: of all closed shapes with the same perimeter, the circle encloses the most area, and among rectangles the square does.

Key facts

• For perimeter P: square area = P² ⁄ 16 and circle area = P² ⁄ 4π.
• At equal perimeters, circle area ÷ square area = 4⁄π ≈ 1.27, whatever the perimeter.
• With 100 m of wire the circle encloses about 795.8 m² and the square 625 m².

Common traps

• Treating the 100 m as a radius or a diameter. It is the circumference, so 2πr = 100 and r ≈ 15.92 m.
• Comparing the circle with the square inscribed in it, which gives π⁄2 ≈ 1.57, instead of with a square of the same perimeter.

No directly related past PYQ was found.