What is the ratio of the area of a regular hexagon to that of an equilateral triangle of the same side length?

Correct — A.
Let the common side be a. Join the hexagon's centre to its six vertices: 6 triangles
Angle at the centre of each: 360° ÷ 6 = 60°
Its two sides from the centre are equal, so the other two angles are (180° − 60°) ÷ 2 = 60° each

Every triangle is equilateral with side a
Hexagon area = 6 × (√3⁄4)a², triangle area = (√3⁄4)a²
Divide, and (√3⁄4)a² cancels: 6 : 1 → option (a)

Why the others are wrong

• (b)
  3:1 — 3 : 1 would make the hexagon three copies of the triangle. Joining its centre to the vertices gives six equilateral triangles of side a, so it holds six copies.
• (c)
  √3 : 1 — √3 : 1 keeps a √3 that cancels. Both areas carry the same (√3⁄4)a² factor, so the ratio is a pure count of triangles: 6.
• (d)
  1:6 — 1 : 6 is the right numbers in the wrong order, triangle to hexagon. The question names the hexagon first, and the hexagon is the larger shape.

A regular hexagon of side a is six equilateral triangles of side a. Its centre is as far from each vertex as the side is long, so the six central triangles have all three sides equal.

So its area is 6 × (√3⁄4)a² = (3√3⁄2)a².

The same picture explains the 120° interior angle: each vertex angle is made of two 60° triangle angles.

Key facts

• Area of an equilateral triangle of side a = (√3⁄4)a²
• Area of a regular hexagon of side a = (3√3⁄2)a², six times that triangle
• In a regular hexagon the distance from the centre to each vertex equals the side

Common traps

• Writing the ratio in reverse, 1 : 6, when the hexagon is named first
• Leaving √3 in the ratio, though it cancels between the two area formulas

No directly related past PYQ was found.