Types of Polygon: Classification with Examples
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 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.