# Hexagon Shape — Definition, Types, Properties, and Area Formula

TL;DR

Hexagon is a six-sided closed two-dimensional polygon with six vertices and six interior angles. In a regular hexagon, all six sides are equal, all six interior angles are 120°, and the sum of interior angles is 720°. The area of a regular hexagon with side s is \( \frac{3\sqrt{3}}{2} s^2 \).

## What Is a Hexagon?

A **hexagon** is a closed two-dimensional polygon with **six sides**, **six vertices**, and **six interior angles**. The name comes from Greek: _hexa_ (six) + _gonía_ (corner / angle) — literally _"six corners"_. A hexagon is a polygon — specifically the one with the smallest number of sides whose interior angles each equal 120° in the regular case. It sits between the pentagon (5 sides) and the heptagon (7 sides) in the polygon family.

## The Five Types of Hexagons

Hexagons are classified by **two independent dimensions**: whether all sides and angles are equal ( _regular_ vs _irregular_) and whether all interior angles are less than 180° ( _convex_ vs _concave_). A complex hexagon — one where sides cross each other — is a separate category.

### 1. Regular Hexagon

All six sides equal, all six interior angles equal (120° each). The most-pictured kind. Has the maximum symmetry possible for a hexagon: six lines of symmetry and rotational symmetry of order 6.

### 2. Irregular Hexagon

A hexagon where the sides are not all equal, or the angles are not all equal (or both). Still has six sides and six interior angles summing to 720°.

### 3. Convex Hexagon

All six interior angles are strictly less than 180°. Every diagonal stays inside the hexagon. A regular hexagon is always convex.

### 4. Concave Hexagon

At least one interior angle is greater than 180° — this creates an _inward dent_ in the shape. At least one diagonal exits the hexagon and re-enters.

### 5. Complex (Self-Intersecting) Hexagon

The sides cross each other — looks like a six-sided star or a tangled hexagon. Less common in school geometry; appears in advanced contexts.

## The Properties of a Hexagon

These properties apply to **all hexagons** (regular and irregular) unless noted.

| Property | Value |
| --- | --- |
| **Number of sides** | 6 |
| **Number of vertices** | 6 |
| **Number of interior angles** | 6 |
| **Sum of interior angles** | 720° |
| **Sum of exterior angles** | 360° (always — true for every polygon) |
| **Number of diagonals** | 9 |
| **Each interior angle (regular only)** | 120° |
| **Each exterior angle (regular only)** | 60° |
| **Lines of symmetry (regular only)** | 6 |
| **Rotational symmetry (regular only)** | Order 6 (60° rotation) |

### Sum of Interior Angles — Why 720°?

The formula for the sum of interior angles of any polygon with n sides:

S = (n−2) × 180°

For a hexagon, n = 6:

S = (6−2) × 180°=4×180°=720°

In a **regular** hexagon, the 720° divides equally across the six angles:

Each interior angle = 720°/6 = 120°.

### Exterior Angles

The exterior angles of _any_ polygon (hexagon or otherwise) sum to exactly 360°. In a regular hexagon, that 360° divides evenly across the six exterior angles:

Each exterior angle = 360°/6 = 60°.

And interior + exterior at each vertex = 120° + 60° = 180° (a linear pair).

### Number of Diagonals

The general formula for the number of diagonals of an n-sided polygon:

D = n(n−3)/2

For a hexagon:

D = 6 × 3/2 = 9 diagonals

These nine diagonals divide a regular hexagon into smaller triangles — six congruent equilateral triangles when you draw all three "long" diagonals through the centre.

## Area of a Hexagon

### Regular Hexagon — The Direct Formula

For a regular hexagon with side length s:

A = \( \frac{3\sqrt{3}}{2} s^2 \)

This is the formula you'll use 90% of the time in school problems.

**Quick numerical reference:**

| Side length s | Area \( \frac{3\sqrt{3}}{2} s^2 \) |
| --- | --- |
| 1 unit | ≈ 2.598 |
| 2 units | ≈ 10.392 |
| 5 units | ≈ 64.952 |
| 10 units | ≈ 259.808 |

### Irregular Hexagon — No Single Formula

For an irregular hexagon, there is **no single area formula**. Three common methods:

- **Decompose into triangles.** Split the hexagon into triangles using diagonals, find each triangle's area, sum.
- **Coordinate (shoelace) formula.** If the six vertices have known (x,y) coordinates, use the shoelace formula:
  \( A = \frac{1}{2} \left| \sum_{i=1}^{n}(x_i y_{i+1} - x_{i+1} y_i) \right| \)

## Perimeter of a Hexagon

### Regular Hexagon

All six sides equal, so:

P = 6s

### Irregular Hexagon

Sum of the six (different) side lengths:

P = a + b + c + d + e + f

### Key Takeaways

- A **hexagon** is a six-sided polygon with six vertices, six interior angles, and an interior-angle sum of 720°.
- A **regular hexagon** has all sides equal, all angles equal (120°), six lines of symmetry, and rotational symmetry of order 6.
- The **area of a regular hexagon** with side s is \( \frac{3\sqrt{3}}{2}s^2 \) .
- A hexagon has **9 diagonals** and **6 exterior angles** (each 60° in a regular hexagon).
- **Hexagons tile the plane efficiently** — the geometric reason they appear in honeycomb, graphene, snowflakes, basalt columns, and bolt heads.