Polygon Shape- Classification and Properties

What Is a Polygon?

A polygon is a closed, flat shape with straight sides. That's it. No curves, no open lines. If you can draw it with straight edges and it forms a complete loop, you're looking at a polygon.

Every polygon has three or more sides and vertices (corner points). Triangles, squares, pentagons—these are all polygons. Circles and shapes with curved edges are not.

Classification of Polygons

Polygons fall into several categories depending on what you're measuring—number of sides, angle type, or regularity.

By Number of Sides

The most basic classification. Mathematicians use Greek prefixes to name polygons based on their side count:

• 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 actual number.

By Angle Type: Convex vs. Concave

Convex polygons have all interior angles pointing outward. No indentations. You can draw a straight line between any two points inside the shape without crossing the boundary.

Concave polygons have at least one interior angle greater than 180°. They look like someone pushed part of the shape inward. A classic example is a star shape.

By Regularity: Regular vs. Irregular

Regular polygons have equal sides and equal angles. A square fits here, so does an equilateral triangle. These shapes are symmetrical and clean.

Irregular polygons have sides and angles that don't match. Most shapes you encounter daily are irregular—rectangles, trapezoids, any random five-sided shape.

Key Properties of Polygons

Interior Angles

Here's the formula for any polygon:

Sum of interior angles = (n − 2) × 180°

Where n is the number of sides.

For a triangle: (3 − 2) × 180° = 180° ✓
For a quadrilateral: (4 − 2) × 180° = 360° ✓
For a hexagon: (6 − 2) × 180° = 720° ✓

If it's a regular polygon, divide that sum by n to get each individual angle.

Exterior Angles

The sum of exterior angles for any polygon is always 360°. This doesn't change regardless of how many sides you have.

Each exterior angle of a regular polygon = 360° ÷ n

Diagonals

A diagonal connects two non-adjacent vertices. The number of diagonals in a polygon:

Diagonals = n(n − 3) ÷ 2

A pentagon has 5(5 − 3) ÷ 2 = 5 diagonals. A hexagon has 9. A decagon has 35.

Polygon Comparison Table

How to Identify and Work With Polygons

Step 1: Count the Sides

Look at your shape. Count every straight edge. That number gives you the basic name—triangle (3), quadrilateral (4), pentagon (5), and so on.

Step 2: Check for Curves

Any curve disqualifies it from being a polygon. The sides must be straight line segments only.

Step 3: Verify It's Closed

Every endpoint must connect to another. No gaps, no open shapes.

Step 4: Classify the Type

Ask yourself two questions:

• Are all sides and angles equal? → Regular polygon
• Does any interior angle point inward? → Concave polygon

Step 5: Calculate What You Need

Use the formulas above depending on what you're solving:

• Need total interior angles? → (n − 2) × 180°
• Need each angle in a regular polygon? → (n − 2) × 180° ÷ n
• Need diagonals? → n(n − 3) ÷ 2

Quick Reference for Common Polygons

Triangle: 3 sides, 180° total interior, 60° per angle if regular
Quadrilateral: 4 sides, 360° total interior, 90° per angle if regular
Pentagon: 5 sides, 540° total interior, 108° per angle if regular
Hexagon: 6 sides, 720° total interior, 120° per angle if regular

The interior angle of any regular polygon approaches 180° as the number of sides increases—but never reaches it. That's why you can't have a two-sided polygon with straight lines.

That's the essentials. Polygons are straightforward once you understand the classification system and can apply the angle formulas. The naming convention, the interior angle sum, and the exterior angle rule cover 90% of what you'll encounter.