An n-gon is simply a polygon with n sides, where “n” stands for any whole number. A triangle is a 3-gon, a square is a 4-gon, a pentagon is a 5-gon, and so on. The term gives mathematicians a convenient shorthand for talking about polygons in general, without specifying a particular number of sides.
The word “polygon” itself comes from the Greek words polús (“many”) and gōnía (“corner” or “angle”). So when you see “n-gon,” you can read it as “n-angled shape” or, more practically, a flat, closed shape with n straight sides.
How Polygons Get Their Names
Every polygon has a name based on its number of sides, drawn from Greek (and occasionally Latin) roots. Here are the most common ones:
- 3 sides: triangle (or trigon)
- 4 sides: quadrilateral (or tetragon)
- 5 sides: pentagon
- 6 sides: hexagon
- 7 sides: heptagon
- 8 sides: octagon
- 9 sides: nonagon (or enneagon)
- 10 sides: decagon
- 12 sides: dodecagon
- 20 sides: icosagon
- 100 sides: hectogon
Once you get past 12 or so sides, the Greek-based names become cumbersome, and nobody wants to say “enneakaidecagon” for a 19-sided shape. That’s exactly why the term “n-gon” exists. Instead of memorizing exotic names, you can just call it a 19-gon and everyone knows what you mean.
Regular vs. Irregular N-Gons
A regular n-gon has all sides the same length and all interior angles equal. A stop sign is a regular octagon (regular 8-gon). An irregular n-gon still has n sides, but the sides and angles can vary freely. Most polygons you encounter in real life, like the shape of a room or a plot of land, are irregular.
Regular n-gons have especially clean symmetry. A regular n-gon has exactly n lines of symmetry. A regular hexagon has 6 lines of symmetry, a regular pentagon has 5, and so on. This symmetry is what makes regular polygons so common in architecture, tiling, and design.
Key Formulas for Any N-Gon
Interior Angle Sum
The interior angles of any n-gon, regular or not, always add up to (n − 2) × 180°. This works because you can divide any polygon into (n − 2) triangles, and each triangle’s angles sum to 180°. A triangle’s angles total 180°. A quadrilateral’s total 360°. A hexagon’s total 720°.
For a regular n-gon, each individual angle equals that total divided by n. So each angle in a regular hexagon is 720° ÷ 6 = 120°.
Number of Diagonals
A diagonal is a line connecting two non-adjacent corners. The total number of diagonals in an n-gon is n(n − 3) / 2. Each vertex connects to (n − 3) other vertices via diagonals (you subtract 3 because a vertex can’t form a diagonal with itself or its two neighbors). A hexagon has 6(3) / 2 = 9 diagonals. A decagon has 35.
Area of a Regular N-Gon
If you know the perimeter (total length of all sides) and the apothem (the distance from the center to the middle of any side), the area is simply apothem × perimeter ÷ 2. The apothem is like the “radius” measured to the flat side rather than to a corner.
As N Gets Larger, You Get a Circle
One of the most elegant ideas in geometry: as n increases, a regular n-gon looks more and more like a circle. A regular 100-gon is nearly indistinguishable from a circle to the naked eye. In the mathematical limit, as n approaches infinity, the regular n-gon becomes a circle. This insight goes back to ancient Greek mathematicians who used polygons with many sides to approximate the circumference and area of circles.
Constructing N-Gons With Compass and Straightedge
A classic question in geometry is which regular n-gons you can draw perfectly using only a compass and an unmarked ruler. Carl Friedrich Gauss and Pierre Wantzel proved that a regular n-gon is constructible this way only if n is a power of 2 (like 4, 8, 16) or a power of 2 multiplied by one or more distinct Fermat primes. Fermat primes are a rare type of prime number with the form 2 raised to a power of 2, plus 1. Only five are known: 3, 5, 17, 257, and 65,537.
This means you can construct a regular triangle (3), square (4), pentagon (5), hexagon (6), octagon (8), and even a regular 17-gon with just a compass and straightedge. But a regular 7-gon or 9-gon? Impossible with those tools alone.
N-Gons in 3D Modeling
The term “n-gon” also shows up in 3D computer graphics, where it has a more specific meaning: a polygon face with more than four vertices. In modeling software, the ideal building blocks are triangles (tris) and four-sided polygons (quads) because they subdivide cleanly and render predictably. N-gons, by contrast, cause problems. They subdivide poorly, disrupt the flow of edges across a surface, and create strange shading artifacts that are difficult to fix.
Some popular sculpting tools like ZBrush and Mudbox will flag n-gons on import, and working across n-gon surfaces produces unreliable results. Models with n-gons also behave unpredictably when moved between different software applications. For these reasons, 3D artists generally treat n-gons as a modeling mistake to clean up rather than a tool to rely on.