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. For the base definition and parts of a polygon, start with polygons.
By the end you will classify any polygon along all four axes and know which properties each label guarantees. The convex-versus-concave split runs deeper in convex polygon; the all-sides-equal case connects to the hexagon and pentagon.
Classification 1: By Number of Sides
The most familiar way to type a polygon is by counting its sides. The Greek-prefix name applies whatever the shape's regularity or convexity.
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 "$n$-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
This axis is about equality of parts, and it is the one students most often half-state.
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 $n$-gon is $\dfrac{(n-2)\times 180°}{n}$.
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.
The trap here is the word "and": a shape with all sides equal is equilateral, and a shape with all angles equal is equiangular, but regular demands both together.
Classification 3: Convex Versus Concave
This axis is about whether any corner caves inward.
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, and some of its diagonals fall outside the figure.
A quick way to test: if you can draw a straight line that crosses the boundary in more than two places, the polygon is concave. The full contrast, with diagonal tests and real-world examples (stars, arrows, the letter L), is the convex-versus-concave distinction linked above.
Classification 4: Simple Versus Complex
A simple polygon has sides that meet only at their shared vertices — its boundary never crosses itself. This is what "polygon" means in everyday geometry.
A complex (or self-intersecting) polygon has sides that cross away from a vertex. A five-pointed star drawn in one stroke (a pentagram) is the classic example. Complex polygons exist, but most school problems assume simple polygons unless they say otherwise.
Axis | Categories | Test |
|---|---|---|
Sides | Triangle … $n$-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 natural first move is to see four equal sides and say "regular." Try it — and notice what is missing: the angles.
A rhombus that is not a square has two acute and two obtuse angles, so its angles are not all equal. Regular requires equal sides and equal angles.
A rhombus is therefore an irregular (equilateral but not equiangular) convex quadrilateral. Only the special rhombus with $90°$ angles — a square — is regular.
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.
At the inner corner of the L, the interior angle is $270°$, a reflex angle greater than $180°$.
So the L is a concave irregular hexagon. The inward notch is exactly what "concave" describes.
Example 4
A regular polygon has each interior angle equal to $135°$. Name the polygon and state its full classification.
Use the regular-polygon angle formula:
$$\frac{(n-2)\times 180°}{n} = 135°$$ $$180n - 360 = 135n$$ $$45n = 360$$ $$n = 8$$
The polygon is a regular octagon — regular, convex (all angles $135° < 180°$), and simple. The eight-sided case is explored in octagon.
Example 5
Is a pentagram (five-pointed star drawn in one continuous stroke) 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. The convex/concave labels apply only to simple polygons, so the question "is it convex?" does not even apply — it is disqualified one step earlier.
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?
A regular hexagon has 6 sides; a stop sign is a regular octagon with 8 sides.
Both are regular, both convex, both simple.
But on the side-count axis they differ: hexagon versus octagon. So they share three of the four labels and differ on one. "Same type" is true for regularity, shape, and structure, false for side count — which is why precise classification needs all four axes, not a single word.
Where Classifying Polygons Earns its Keep
Sorting polygons by type is not busywork — the type decides which formulas and properties you are allowed to use.
Formulas depend on regularity. The clean area formula $A = \tfrac{1}{2},P,a$ (half perimeter times apothem) works only for regular polygons, because only regular polygons have a single apothem; an irregular polygon has no such center distance. Misjudge the type and you reach for a formula that does not apply.
Algorithms depend on convexity. In computer graphics and robotics, "is this region convex?" decides which collision-detection and path-planning methods run. Convex regions are fast to test; concave ones must first be broken into convex pieces.
Why the four axes are independent. The destination is precision: each axis answers a different question, so collapsing them into one "type" loses information you will need the moment you compute an area or run an algorithm.
The convex-decomposition problem — cutting a concave polygon into the fewest convex pieces — is a studied question in computational geometry, and it exists precisely because so many real-world outlines are concave.
Mistakes To Watch For
Mistake 1: Treating "equilateral" as "regular"
Where it slips in: Seeing equal sides and declaring the polygon regular without checking the angles.
Don't do this: Call a rhombus regular because its four sides match.
The correct way: Confirm both equal sides and equal angles. A rhombus is equilateral but not equiangular, so it is irregular. The student who reads "regular" as "looks neat" makes this slip on every non-square rhombus and kite.
Mistake 2: Applying convex-only properties to a concave polygon
Where it slips in: Assuming all diagonals stay inside, or that the interior angles all sit below $180°$, before checking for a reflex angle.
Don't do this: Use a convex-only area or diagonal argument on an arrow-shaped polygon.
The correct way: Scan for a reflex angle first. One interior angle above $180°$ makes the polygon concave, and the convex shortcuts no longer hold. The student who pictures every polygon as a tidy regular shape forgets that concave ones even exist.
Mistake 3: Forgetting the simple-versus-complex check
Where it slips in: Classifying a self-crossing figure as convex or concave when it is neither.
The correct way: Check that the polygon is simple before asking convex or concave — those labels only apply to non-self-intersecting figures.
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.
A Practical Next Step
Work through the exercises below. For each shape, record all four classification labels before moving on.
Classify a non-square rectangle along all four axes. (Answer to Question 1: quadrilateral, irregular, convex, simple.)
A regular polygon has each interior angle equal to $156°$. Name it. (Answer to Question 2: $n = 15$, a regular pentadecagon.)
To go deeper into classifying shapes with a teacher, explore Bhanzu's geometry tutor, our middle school math tutor sessions, or math tutoring options. To classify polygons live with a trainer, you can book a free demo class.
Read More
Interior angles — the angles the regular-or-irregular test relies on.
Exterior angles of a polygon — the angles that always sum to 360°.
Quadrilaterals — the four-sided polygons, where concavity first becomes possible.
Types of triangles — the three-sided polygon classified by sides and angles.
Geometric shapes — the broader 2-D and 3-D shape family.
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