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.

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:

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.