# Hexagon

A hexagon is a 6-sided geometric figure commonly found in everyday life. Some hexagons in real life include honeycomb and the nut that a threaded bolt is screwed into. Each cell that makes up a honeycomb is in the shape of a hexagon.

## What is a hexagon

A hexagon is a two-dimensional polygon with 6 sides and 6 angles. The prefix "hexa" denotes the number 6. Below are three hexagon examples.

The hexagon on the left is a regular hexagon and is likely what most people think of when they hear the word hexagon. The other two hexagons are irregular, and the right-most one can further be categorized as a concave hexagon. As we can see, there are many different types of hexagons, and we categorize them based on their characteristics.

## Types of hexagons

Hexagons are categorized as either regular or irregular and concave or convex.

### Regular hexagon

A regular hexagon is one in which all of the sides and interior angles of the hexagon are equal. A regular hexagon is always convex. The figure below shows a regular hexagon example:

### Irregular hexagon

An irregular hexagon is one in which not all of the sides and interior angles are equal. The figure below shows an irregular hexagon example:

### Convex hexagon

A convex hexagon is one in which all of the interior angles measure less than 180°. A regular hexagon is always convex, but a hexagon does not have to be regular in order to be convex. The figure below shows an irregular convex hexagon example:

Notice that the lengths of the sides, and therefore the measures of the interior angles, are not all equal. Also, none of the interior angles is larger than 180°, so the hexagon is convex.

### Concave hexagon

A concave hexagon is a hexagon in which at least one of the interior angles has a measure greater than 180°. A concave hexagon cannot be a regular hexagon; it is always an irregular hexagon. The figure below shows a concave hexagon example:

Since one of the interior angles of the hexagon has a measure greater than 180°, the hexagon is a concave hexagon.

## Hexagon properties

The following are various properties of all hexagons as well as properties of a regular hexagon.

### Properties of all hexagons

- All hexagons are two-dimensional closed figures with 6 edges, faces, and vertices.
- The sum of the internal angles of all hexagons is equal to 720°.
- All hexagons have 9 diagonals

### Properties of regular hexagons

- The sides and interior angles of a regular hexagon are all equal.
- The interior angles of a regular hexagon measure 120°.
- The exterior angles of a regular hexagon measure 60°.
- The sum of the interior angles of a regular hexagon is 720°.
- A regular hexagon is always convex.
- A regular hexagon can be divided into 6 equilateral triangles.
- The opposite sides of a regular hexagon are parallel to each other.
- A regular hexagon has 6 lines of symmetry.
- A regular hexagon has 6 60° angles of rotation. This means that it has a rotational symmetry of order 6 and can be rotated in such a way that it will look the same as the original shape 6 times in 360°.

## Regular hexagon area formula

The formula for the area of a regular hexagon with side length s is:

From the center, a regular hexagon can be divided into six equilateral triangles, each having side length, s, as shown below.

The area, T, of one of the equilateral triangles, drawn in blue, can be found by using , where the apothem is the height of the triangle. Since there are six equilateral triangles, the area of a regular hexagon is .

Example:

Find the area of a regular hexagon that has a side length of 8.

Using :

### Regular hexagon area using apothem

To find the area of a regular hexagon using its apothem, we need to know the perimeter of the hexagon and be able to find its apothem. Given that we know these measures, the formula for the area of a regular hexagon using its apothem is,

where P is the perimeter and a is the apothem of the hexagon represented by the dotted red line in the figure below:

## Hexagon perimeter

The formula for the perimeter of a regular hexagon with side length s is:

To find the perimeter of an irregular hexagon, find the sum of the lengths of the sides of the hexagon. Depending on the hexagon and the provided information, this may involve breaking the hexagon up into triangles and using a variety of triangle formulas to find the side lengths of the hexagon.

## Hexagon diagonals

A hexagon has 9 diagonals. A regular hexagon has 6 short diagonals and 3 long diagonals. The diagonals of a hexagon are line segments joining two non-consecutive vertices. Three diagonals can be drawn from each vertex for a total of nine diagonals in any hexagon.

- Long diagonal - The long diagonal of a regular hexagon is one of the 3 diagonals that crosses the middle of a regular hexagon.
- Short diagonal - The short diagonal of a regular hexagon is one of the 6 diagonals that do not cross the middle of a regular hexagon.

The figure below shows the diagonals of a regular hexagon.

The diagonals shown in red are the long diagonals and the diagonals shown in blue are the short diagonals. Note that long diagonals and short diagonals apply to regular hexagons. The figure below shows an example of an irregular convex hexagon and its diagonals.

The formula for the number of diagonals, d_{n}, in any polygon is,

where n is the number of sides in the polygon. Since a hexagon has 6 sides:

For a regular hexagon, the length of the long diagonal is , where s is the length of the side of the hexagon; the length of the short diagonal is where a is the apothem. Refer to the figure below.

## Interior angles of a hexagon

The sum of the interior angles of a hexagon equals 720°.

As shown in the figure above, three diagonals can be drawn to divide the hexagon into four triangles. The blue lines above show just one way to divide the hexagon into triangles; there are others. The sum of interior angles of the four triangles equals the sum of interior angles of the hexagon. Since the sum of the interior angles of a triangle is 180°, the sum of the interior angles of the hexagon is 4 × 180° = 720°.