# 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

| Polygon | Sides | Vertices | Interior angle sum |
| --- | --- | --- | --- |
| Triangle | 3 | 3 | 180° |
| Quadrilateral | 4 | 4 | 360° |
| Pentagon | 5 | 5 | 540° |
| Hexagon | 6 | 6 | 720° |
| Heptagon | 7 | 7 | 900° |
| Octagon | 8 | 8 | 1080° |
| Nonagon | 9 | 9 | 1260° |
| Decagon | 10 | 10 | 1440° |
| n-gon | n | n | \( (n−2)\times180° \) |

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.