Equilateral Polygon — Definition, Formula & Examples

An equilateral polygon is a polygon in which all sides have the same length. A rhombus and a square are both equilateral quadrilaterals, but only the square is also equiangular.

A polygon with $n$ sides is equilateral if and only if all $n$ of its side lengths are congruent, i.e., $s_1 = s_2 = \cdots = s_n$ . No constraint is placed on its interior angles.

How It Works

To determine whether a polygon is equilateral, measure or calculate every side length. If all sides are equal, the polygon is equilateral regardless of its angles. An equilateral polygon that is also equiangular (all angles equal) is called a regular polygon. A polygon can be equilateral without being regular — for example, a rhombus has four equal sides but its angles are not all 90°.

Worked Example

Problem: A quadrilateral has vertices at A(0, 0), B(4, 0), C(6, 4), and D(2, 4). Is it equilateral?

Step 1: Find each side length using the distance formula.

AB = \sqrt{(4-0)^2 + (0-0)^2} = 4

Step 2: Compute the remaining sides.

BC = \sqrt{(6-4)^2 + (4-0)^2} = \sqrt{4+16} = \sqrt{20}

Step 3: Since AB = 4 and BC = √20 ≈ 4.47, the sides are not all equal.

Answer: No. Because

AB \neq BC

, this quadrilateral is not equilateral (it is a parallelogram but not a rhombus).

Why It Matters

Understanding equilateral polygons helps you classify shapes precisely in geometry courses. In tiling and design problems, equilateral polygons (like rhombuses and equilateral triangles) appear frequently because equal side lengths create predictable, repeating patterns.

Common Mistakes

Mistake: Assuming an equilateral polygon must also be equiangular (regular).

Correction: Equilateral only means equal sides. A rhombus is equilateral but not equiangular. A polygon must be both equilateral and equiangular to be regular.