A polygon is regular when it meets two conditions at once: all of its sides are the same length, and all of its interior angles are the same size. Miss either condition and the polygon is irregular. A rectangle, for example, has four equal angles but its sides come in two different lengths, so it doesn’t qualify. A rhombus has four equal sides but its angles aren’t all equal, so it doesn’t qualify either. Only a square checks both boxes, making it the sole regular quadrilateral.
The Two Rules: Equal Sides and Equal Angles
Geometers use two terms for these properties. “Equilateral” means all sides are congruent, and “equiangular” means all angles are congruent. A regular polygon is both equilateral and equiangular. For triangles, having one property automatically guarantees the other: if a triangle has three equal sides, it must have three equal angles. But for shapes with four or more sides, the two properties are independent. You need to verify both.
This is why a stop sign (a regular octagon) looks so balanced. Each of its eight sides is the same length, and each interior angle measures exactly 135 degrees. Compare that to a random eight-sided shape where the sides and angles vary wildly, and you can see why regularity is a distinct geometric category.
How to Find the Interior Angle
Every regular polygon has a predictable interior angle based on the number of sides. The formula is straightforward: subtract the fraction 360 divided by the number of sides from 180. In other words, interior angle = 180 − (360 ÷ n), where n is the number of sides.
Here’s what that produces for common regular polygons:
- Equilateral triangle (3 sides): 60°
- Square (4 sides): 90°
- Regular pentagon (5 sides): 108°
- Regular hexagon (6 sides): 120°
- Regular octagon (8 sides): 135°
- Regular decagon (10 sides): 144°
- Regular dodecagon (12 sides): 150°
Notice the pattern: as the number of sides increases, the interior angle gets closer and closer to 180°. A regular polygon with hundreds of sides would have angles so close to 180° that it would look virtually identical to a circle.
The Exterior Angle Shortcut
There’s an even simpler way to think about regular polygon angles. The exterior angles of any polygon, regular or not, always add up to exactly 360°. For a regular polygon, every exterior angle is the same size, so you just divide: exterior angle = 360° ÷ number of sides. A regular hexagon has exterior angles of 60°. A regular octagon has exterior angles of 45°.
Since each interior angle and its corresponding exterior angle sit on a straight line, they always add up to 180°. So once you know one, you know the other. Many geometry students find it easier to start with the exterior angle and subtract from 180 to get the interior angle, rather than memorizing a separate formula.
Symmetry of Regular Polygons
Regular polygons are the most symmetrical flat shapes possible. A regular polygon with n sides has exactly n lines of symmetry and rotational symmetry of order n. A regular pentagon, for instance, has five lines of symmetry and looks identical after being rotated by any multiple of 72° (which is 360 ÷ 5). A regular hexagon has six lines of symmetry and looks the same every 60° of rotation.
This is actually another way to test for regularity. If a polygon doesn’t have as many lines of symmetry as it has sides, it isn’t regular. A rectangle has only two lines of symmetry despite having four sides, which confirms it falls short of regularity.
Names for Regular Polygons by Side Count
Regular polygons are named by their number of sides, using the same names as irregular polygons. The “regular” label just gets added in front:
- 3 sides: equilateral triangle
- 4 sides: square
- 5 sides: regular pentagon
- 6 sides: regular hexagon
- 7 sides: regular heptagon
- 8 sides: regular octagon
- 9 sides: regular nonagon
- 10 sides: regular decagon
- 11 sides: regular hendecagon
- 12 sides: regular dodecagon
The equilateral triangle and the square are special cases. They’re so common that we rarely bother saying “regular triangle” or “regular quadrilateral,” though both terms are technically correct.
Regular Polygons in the Real World
Honeycomb cells are regular hexagons, a shape bees use because it tiles a flat surface without gaps while enclosing the most area per unit of wax. Snowflakes exhibit six-fold symmetry rooted in the hexagonal structure of ice crystals, and basalt columns (like those at Giant’s Causeway in Ireland) often cool into hexagonal cross-sections for similar structural reasons.
In architecture and design, regular polygons show up constantly. Triangular windows appear in churches, sometimes representing the trinity. Hexagonal tile patterns date back to ancient Pompeii, where designers combined regular hexagons with squares and equilateral triangles in elaborate floor mosaics. Modern stop signs are regular octagons, chosen partly because the distinctive shape is recognizable even when covered in snow or grime.
What About Star Polygons?
Regular polygons are typically convex, meaning no interior angle exceeds 180° and no sides cross each other. But regularity can extend to star shapes as well. A five-pointed star drawn with equal side lengths and equal angles at every point is considered a regular star polygon. These shapes follow the same core principle: identical sides, identical angles. The difference is that their sides intersect, creating the familiar star pattern rather than a simple closed outline. A standard five-pointed star and the Star of David (built from two overlapping triangles) are common examples of this broader category.