Classifying Polygons- Complete Guide

What Is a Polygon?

A polygon is a flat, two-dimensional shape with straight sides that form a closed loop. That's the basic definition. No curves, no open lines, no exceptions.

Every polygon has:

• At least 3 sides
• Vertices (corners) where sides meet
• Interior angles that add up to a specific value

Triangles, squares, pentagons — they're all polygons. Circles are not. Neither are shapes with curved edges or gaps. Those are just shapes, not polygons.

Classifying Polygons by Number of Sides

The most basic classification system counts the sides. Each polygon has a specific name based on how many sides it has.

• 3 sides = Triangle
• 4 sides = Quadrilateral
• 5 sides = Pentagon
• 6 sides = Hexagon
• 7 sides = Heptagon
• 8 sides = Octagon
• 9 sides = Nonagon
• 10 sides = Decagon
• 12 sides = Dodecagon

Anything with more than 12 sides typically gets called an "n-gon" where n equals the number of sides. A 15-sided shape is a 15-gon. A 47-sided shape is a 47-gon.

Regular vs Irregular Polygons

This classification depends on whether all sides and angles are equal.

Regular Polygons

In a regular polygon, all sides are the same length and all interior angles are equal. A square is regular. An equilateral triangle is regular. A regular pentagon has five identical sides and five identical angles.

The more sides a regular polygon has, the closer it gets to looking like a circle. A regular 100-gon looks almost indistinguishable from a circle at normal sizes.

Irregular Polygons

An irregular polygon has sides of different lengths or angles of different measures. A rectangle is irregular — opposite sides are equal, but adjacent sides are different lengths. Most shapes you see in real life are irregular polygons.

Irregular doesn't mean random. It just means the shape lacks the symmetry of a regular polygon.

Convex vs Concave Polygons

This classification depends on the shape's interior angles and overall geometry.

Convex Polygons

A convex polygon has no interior angle greater than 180 degrees. If you pick any two points inside the shape, the line connecting them stays entirely inside the shape.

Triangles, squares, regular pentagons — all convex. You can think of convex as "bulging outward" or having no indentations.

Concave Polygons

A concave polygon has at least one interior angle greater than 180 degrees. It has at least one indentation — a "cave" in the shape.

A dart is a concave quadrilateral. A star shape is a concave polygon. If you can draw a line between two points inside the shape that passes through the outside, you're looking at a concave polygon.

Simple vs Complex Polygons

This classification deals with whether the polygon's edges cross each other.

Simple Polygons

A simple polygon has edges that only meet at vertices. No crossing lines, no self-intersection. Every side connects cleanly to the next. This is what most people mean when they say "polygon."

Complex Polygons

A complex polygon has edges that cross over each other. A star of David is a complex hexagon — two triangles overlapping. A pentagram is a complex polygon with crossing lines.

Some definitions of "polygon" exclude complex polygons entirely. If you're working on geometry problems, check whether your context allows them.

How to Classify Any Polygon

Follow this step-by-step process to classify any polygon you encounter.

Step 1: Count the Sides

Start here. Count every straight edge that forms the boundary. A square has 4. A hexagon has 6. Name the shape accordingly.

Step 2: Check for Self-Intersection

Are any edges crossing over each other? If yes, it's a complex polygon. If no, proceed to step 3.

Step 3: Look for Indentations

Does the shape have any inward "dents"? If any interior angle exceeds 180 degrees, it's concave. Otherwise, it's convex.

Step 4: Compare Sides and Angles

Are all sides equal length? Are all angles equal measure? If both are yes, it's regular. If either is no, it's irregular.

That's it. Four steps. A pentagon can be regular convex, irregular convex, regular concave, or irregular concave. Same four steps apply to any polygon.

Polygon Classification Table

The interior angle sum formula: (n - 2) × 180°, where n equals the number of sides. A 20-gon has an interior sum of (20 - 2) × 180 = 3,240°.

Common Mistakes to Avoid

Confusing quadrilaterals with squares. Every square is a quadrilateral, but not every quadrilateral is a square. Rectangles, parallelograms, trapezoids — they're all quadrilaterals.

Forgetting that concave polygons can still be regular. A regular star shape? Actually exists. The interior angles are still equal, even with the indentation. Wait — no, that's wrong. A truly regular polygon cannot be concave. Regularity requires convexity. A star is complex, not regular.

Misidentifying complex polygons. Students often call a star a "concave polygon." It's not. It's complex. The edges cross. Different classification.

Thinking polygons must be symmetrical. Most polygons are irregular. Regular polygons are the exception, not the rule. Stop assuming every shape follows a pattern.

Why This Matters

Polygon classification isn't abstract math nonsense. Architects use polygon properties to design stable structures. Game developers classify collision boundaries as polygons for physics calculations. Computer graphics render everything as polygons — the more polygons, the smoother the image.

Understanding the differences between convex and concave affects path-finding algorithms. Knowing the difference between simple and complex polygons determines whether your shape can be filled with color or must be rendered as an outline.

It's basic geometry. But basic geometry runs the world.