What is Polygon — Definition, Types, Properties & Examples

A polygon is a closed, two-dimensional figure made entirely of straight line segments — no curves, no open ends. This article gives the formal definition, walks through classification by number of sides (triangle → decagon), regular vs irregular, convex vs concave, the interior-angle-sum formula (n−2)×180°, three worked examples (Quick / Standard / Stretch), and the most common mistakes.

A polygon is a closed plane figure built from straight line segments that meet only at their endpoints. The simplest is the triangle (3 sides); the family extends to four-sided, five-sided, all the way up to "n-gon" for any positive integer n ≥ 3.

The Formal Definition

A polygon is a closed two-dimensional figure formed by a finite number of straight line segments (called sides) that intersect only at their endpoints (called vertices). For a shape to count as a polygon:

• It must be closed — the sides form a continuous loop with no gaps.
• It must lie in a single plane — every side is in the same flat surface.
• The sides must be straight — no curves; a circle is not a polygon.
• The sides must meet only at the vertices — they cannot cross or overlap.

A polygon with n sides has n vertices, n interior angles, and ( \frac{n(n−3)}{2} ) diagonals.

Quick reference.

• Definition: closed plane figure with straight sides only.
• Sides === Vertices === Interior angles = n = n = n.
• Sum of interior angles: ( (n−2) \times 180° ).
• Sum of exterior angles: 360° for any polygon (regular or irregular).
• Each interior angle (regular): ( \frac{(n−2) \times 180°}{n} ).
• Each exterior angle (regular): ( \frac{360°}{n} ).
• Number of diagonals: ( \frac{n(n−3)}{2} ).

Classification by Number of Sides

Each new side adds 180° to the interior angle sum — a clean linear pattern.

Classification by Regularity and Shape

Polygons fall into four main categories by shape.

• Regular polygon. All sides equal and all interior angles equal.
• Irregular polygon. Sides and/or angles not all equal.
• Convex polygon. Every interior angle is less than 180°.
• Concave polygon. At least one interior angle is greater than 180°.

A polygon can be both irregular and convex (a scalene triangle); regular shapes are always convex.

Three Properties of Polygon

Interior angle sum

For any polygon with n sides:

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

Exterior angle sum

For any convex polygon — regardless of n:

Sum of exterior angles = 360°.

Number of diagonals

From any vertex, you can draw a diagonal to all other vertices except itself and its two adjacent neighbors — giving (n−3) diagonals per vertex. Dividing by 2 to remove double-counting:

Number of diagonals = ( \frac{n(n−3)}{2} ).

Three Worked Examples of Polygon — Quick, Standard, Stretch

Quick. What is the sum of the interior angles of a heptagon (n=7)?

Apply (n−2) × 180° with n=7:

(7−2) × 180° = 5 × 180° = 900°.

Final answer: 900°.

Standard. Each interior angle of a regular polygon is 144°. How many sides does it have?

The right approach uses the regular-polygon angle formula:

Set this equal to 144°:

( \frac{(n−2) \times 180°}{n} = 144° )

Cross-multiply: (n−2) × 180° = 144°n.

So n=10.

Final answer: 10 sides — a regular decagon.

Stretch. Find the number of sides of a polygon whose interior angles sum to 1440°.

Use (n−2) × 180° = 1440°. Solve: n−2=8, so n=10.

Final answer: 10 sides — a decagon.

Where Polygons Appear — From Honeycomb to Stop Signs

• Honeycombs. Bees build their wax cells as regular hexagons.
• Stop signs. A regular octagon.
• Computer graphics. Every 3D model in a video game is a mesh of triangles.
• City planning. The pentagon-shaped Pentagon building.
• Architecture. Hexagonal floor tiles, octagonal towers, and pentagonal domes use polygons.

Tripping Points to Avoid in Polygon

Mistake 1: Calling a circle a polygon

A polygon has straight sides only. A circle has a single curved edge.

Mistake 2: Counting an open shape as a polygon

A polygon must be closed — start and end at the same vertex with no gap.

Mistake 3: Using the regular-polygon angle formula on an irregular polygon

A correct way: Use the angle-sum to find the missing angle by subtraction.

Conclusion

• A polygon is a closed two-dimensional shape made of straight line segments.
• Polygons classify by side count, regularity, and angle shape.
• Use the regular-polygon formulas only when the polygon is genuinely regular; for irregular ones, work with the sum and subtract.

Three Problems to Cement Polygons

1. Find the sum of interior angles of an octagon.
2. Each interior angle of a regular polygon is 135°. How many sides does it have?
3. A pentagon has angles 100°,110°,115°,90°. Find the fifth angle.