# Types of Polygon: Classification with Examples

Polygons are classified along four independent axes: by number of sides, by regularity, by shape, and by structure. This article works through every type with properties, a comparison table, examples, and the classification mistakes to avoid.

A **polygon** is a closed, flat figure made of straight sides; its _types_ come from four separate classification systems applied together. By **side count** a polygon takes a name (a 5-gon is a pentagon). By **regularity** it is either _regular_ (all sides and angles equal) or _irregular_. By **shape** it is either _convex_ (no interior angle past 180°) or _concave_. By **structure** it is _simple_ (sides meet only at vertices) or _complex_ (sides cross). Every polygon answers all four.

## Classification 1: By Number of Sides

The most familiar way to type a polygon is by counting its sides.

| Sides | Name | Sides | Name |
| --- | --- | --- | --- |
| 3 | Triangle | 8 | Octagon |
| 4 | Quadrilateral | 9 | Nonagon |
| 5 | Pentagon | 10 | Decagon |
| 6 | Hexagon | 11 | Hendecagon |
| 7 | Heptagon | 12 | Dodecagon |

For a side count without a common name, write "nnn-gon" — a 17-sided polygon is simply a 17-gon. The number of vertices always equals the number of sides.

## Classification 2: Regular Versus Irregular

A **regular polygon** satisfies _both_ conditions: all sides are equal in length **and** all interior angles are equal. A square is regular; so is an equilateral triangle and a regular hexagon. Each interior angle of a regular nnn-gon is (n−2)×180°/n.

An [**irregular polygon**](/content/math/geometry/irregular-polygons/index.html) fails at least one condition. A rectangle that is not a square is irregular (equal angles, unequal sides). A rhombus is irregular (equal sides, unequal angles). Most polygons in the real world are irregular.

## Classification 3: Convex Versus Concave

A **convex polygon** has every interior angle strictly less than 180°. No corner dents in, and every diagonal stays inside the figure. A triangle is always convex; every regular polygon is convex.

A **concave polygon** (also called non-convex) has at least one **reflex** interior angle, greater than 180°, so part of the boundary pushes inward, like a notch or an arrowhead. A concave polygon must have at least four sides.

## Classification 4: Simple Versus Complex

A **simple polygon** has sides that meet only at their shared vertices — its boundary never crosses itself. A **complex** polygon has sides that cross away from a vertex. A five-pointed star drawn in one stroke (a pentagram) is the classic example.

| Axis | Categories | Test |
| --- | --- | --- |
| Sides | Triangle … nnn-gon | Count the sides |
| Regularity | Regular / Irregular | All sides AND all angles equal? |
| Shape | Convex / Concave | Any interior angle > 180°? |
| Structure | Simple / Complex | Do any sides cross? |

## Examples of Types of Polygon
### Example 1
**Classify a square along all four axes.**
By sides: 4 sides, so a quadrilateral.
By regularity: all sides equal and all angles 90°, so regular.
By shape: every angle is 90°<180°, so convex.
By structure: no sides cross, so simple.
A square is a regular, convex, simple quadrilateral.

### Example 2
**A student calls a rhombus a regular polygon because "all four sides are equal." Correct the classification.**
A rhombus that is not a square has two acute and two obtuse angles, so its angles are _not_ all equal. Therefore it is an **irregular** convex quadrilateral.

### Example 3
**Classify the shape of the capital letter "L" (six straight sides forming a right-angle notch).**
The L has 6 straight sides meeting at vertices, so it is a hexagon by side count. Its sides are not all equal and its angles are not all equal, so it is irregular. The interior angle is 270°, a reflex angle greater than 180°. So the L is a **concave** irregular hexagon.

### Example 4
**A regular polygon has each interior angle equal to 135°. Name the polygon and state its full classification.**
The polygon is a regular octagon — regular, convex, and simple.

### Example 5
**Is a pentagram a simple or complex polygon, and is it convex?**
The pentagram's sides cross one another away from the vertices. A boundary that crosses itself is self-intersecting. So the pentagram is a **complex** polygon, not simple.

### Example 6
**A garden bed is shaped like a regular hexagon. A landscaper claims it is "the same type" as a stop sign. Are they the same type?**
Both are regular, both convex, both simple. But on the side-count axis they differ: hexagon versus octagon.

## Key Takeaways
- Every polygon carries four labels at once: side count, regularity, shape, and structure.
- **Regular** means equal sides AND equal angles; missing either makes it **irregular**.
- **Convex** means no interior angle exceeds 180°; a single reflex angle makes it **concave**.
- **Simple** polygons never self-cross; convex and concave labels apply only to simple polygons.
- The type you assign decides which formulas and algorithms you are allowed to use.