A regular hexagon has a perimeter of 72 cm. What is its area?

Correct — A. A regular hexagon is six equilateral triangles of the same side.

Side = perimeter ÷ 6 = 72 ÷ 6 = 12 cm
One triangle = (√3⁄4) × 12² = 36√3 cm²
Hexagon = 6 × 36√3 = 216√3 cm²

Area ≈ 216 × 1.73205 ≈ 374.12 cm² → option (a)

Why the others are wrong

• (b)
  449.76 cm² — 449.76 cm² is the area of a regular hexagon of side about 13.2 cm, whose perimeter is about 79 cm, not 72 cm.
• (c)
  670.32 cm² — 670.32 cm² belongs to a hexagon of side about 16.1 cm, with a perimeter near 96 cm. It is far too large for a 72 cm perimeter.
• (d)
  748.15 cm² — 748.15 cm² is roughly double the true area. It is close to 3√3 × 12² ≈ 748.25, what you get if the ½ in (3√3⁄2)s² is dropped.

Joining the centre of a regular hexagon to its six corners cuts it into six equilateral triangles, each with the hexagon's side as its side.

So the area is 6 × (√3⁄4)s² = (3√3⁄2)s², about 2.598s².

The same picture explains why a regular hexagon inscribed in a circle has its side equal to the radius: each triangle's other two sides are radii.

Key facts

• Area of a regular hexagon = (3√3⁄2)s² ≈ 2.598s².
• Area of an equilateral triangle = (√3⁄4)s².
• Perimeter of a regular hexagon = 6s.
• √3 ≈ 1.732.

Common traps

• Using the perimeter, 72, as the side: (3√3⁄2) × 72² ≈ 13,468 cm², nowhere near the options.
• Dropping the ½ in (3√3⁄2)s², which roughly doubles the answer.

No directly related past PYQ was found.