An equilateral polygon is any polygon whose sides are all the same length. That single rule is the only requirement. The angles between those sides can vary, which means equilateral polygons come in a surprisingly wide range of shapes, from perfectly symmetrical figures like squares to lopsided, concave forms that barely look regular at all.
The One Rule: Equal Side Lengths
The word “equilateral” breaks down simply: “equi” means equal, “lateral” means side. So an equilateral polygon is a flat, closed shape where every edge measures the same. A triangle with three 4-inch sides qualifies. So does a diamond (rhombus) with four 6-centimeter sides, even if its angles are sharply skewed.
This is a looser category than most people expect. When you picture a “perfect” shape like a square or a regular hexagon, you’re thinking of a regular polygon, which is a stricter classification. Equilateral polygons only care about side length. The angles can do whatever they want, as long as the shape still closes.
Equilateral vs. Equiangular vs. Regular
These three terms describe overlapping but distinct properties, and mixing them up is one of the most common mistakes in basic geometry.
- Equilateral means all sides are the same length. Angles can differ.
- Equiangular means all interior angles are the same size. Side lengths can differ.
- Regular means both: all sides equal and all angles equal.
A square is a regular quadrilateral because it has four equal sides and four 90-degree angles. A rhombus is equilateral (four equal sides) but not necessarily equiangular, since its angles can be 60° and 120° instead of all 90°. A rectangle is equiangular (four 90° angles) but not equilateral unless it happens to also be a square.
Here’s the one exception that trips people up: for triangles, equilateral automatically means equiangular. If all three sides of a triangle are equal, the angles must each be 60°. This makes every equilateral triangle a regular polygon by default. That shortcut only works for three-sided figures. Once you have four or more sides, equal lengths no longer guarantee equal angles.
Common Examples
The equilateral triangle is the simplest and most familiar example. Its three equal sides force three equal 60° angles, giving it perfect symmetry. It’s the building block for countless structures in engineering and design.
The rhombus is the go-to example for showing that equilateral doesn’t mean regular. All four sides of a rhombus are the same length, but unless every angle is 90° (making it a square), the shape leans to one side like a tilted box. There are infinitely many rhombi with the same side length, each with a different pair of angles. Knowing the side length alone tells you nothing about the shape’s proportions.
Regular pentagons, hexagons, and octagons (think stop signs) are all equilateral, but they’re also equiangular, putting them in the regular polygon category. You can, however, construct equilateral pentagons and hexagons with unequal angles. These look irregular and asymmetric even though every side measures the same.
Concave and Self-Intersecting Shapes
Once a polygon has four or more sides, being equilateral doesn’t even require the shape to be convex. A concave equilateral polygon has at least one interior angle greater than 180°, creating a “dent” in the outline. Picture a five-pointed star shape where every segment is the same length but some corners point inward. The sides are equal, the shape is closed, and it still qualifies as equilateral.
Self-intersecting equilateral polygons also exist. These are shapes where the edges cross over each other, like a star polygon. They push the definition to its limits, but the core rule holds: if every edge is the same length, the polygon is equilateral regardless of how strange it looks.
Angles in Equilateral Polygons
For any convex polygon with n sides, the interior angles always add up to (n − 2) × 180°. A triangle’s angles sum to 180°. A quadrilateral’s sum to 360°. A pentagon’s sum to 540°. This rule applies whether the polygon is equilateral or not.
In a regular polygon, you simply divide that total evenly among all angles. Each angle of a regular hexagon, for instance, is 720° ÷ 6 = 120°. But in an equilateral polygon that isn’t regular, the individual angles can be different sizes as long as they still hit that same total. A rhombus might have two angles of 50° and two of 130°, which adds up to 360° just like a square’s four 90° angles do.
For concave equilateral polygons, the picture gets more complex. With pentagons and higher shapes, allowing concave angles means that knowing just one or two angles is no longer enough to pin down the full shape. The number of possible configurations grows quickly.
Perimeter and Practical Calculations
The one calculation that’s always simple for equilateral polygons is perimeter. Since every side is the same length, the perimeter is just the side length multiplied by the number of sides: P = n × s. An equilateral pentagon with 3-centimeter sides has a perimeter of 15 centimeters, no matter how irregular its angles are.
Area is a different story. For regular polygons, standard formulas exist because the angles are predictable. For equilateral polygons that aren’t regular, you need additional information (specific angle measurements or diagonal lengths) to calculate area, because the same set of equal sides can enclose very different amounts of space depending on how the angles are arranged.
A Useful Geometric Property
One elegant property of equilateral polygons involves distances to the sides. Viviani’s theorem, originally proven for equilateral triangles, states that if you pick any point inside an equilateral triangle and measure the shortest distance from that point to each of the three sides, those three distances always add up to the same total, no matter where the point is. That total equals the height of the triangle.
This property generalizes to all convex equilateral polygons. For any convex polygon with equal side lengths, the sum of the perpendicular distances from any interior point to all the sides remains constant. It’s a surprising result: even if the polygon’s angles are unequal, the equal side lengths alone are enough to guarantee this consistency.