The hexagon is defined as a polygon that has six sides and six interior angles. The interior angles of a hexagon add up to 720°. Taking different characteristics, it is possible to distinguish some types of hexagons. For example, depending on the length of the sides, we can have regular and irregular hexagons, and depending on the contour, we can have convex and concave hexagons.

Here, we will look at a definition of hexagons and a description of the most relevant types of hexagons. In addition, we will learn about some of the fundamental characteristics of these geometric figures. Finally, we will get to know its most important formulas and use them to solve some regular hexagon problems.

## Definition of a hexagon

A hexagon is defined as a polygon with 6 sides and 6 interior angles. Recall that a polygon is a closed two-dimensional (2D) figure that is made up of straight segments. All the sides of the hexagon meet each other end to end to form a shape.

Taking into account the different characteristics that hexagons can have, we can distinguish the following types of hexagons:

- Regular and irregular
- Convex and concave

### Irregular and regular hexagons

A regular hexagon is characterized by having sides with the same length and angles with the same measure. On the other hand, irregular hexagons have sides with different lengths and angles of different measures.

### Convex and concave hexagons

A convex hexagon is a geometric figure in which all its vertices are pointing outward. On the other hand, a concave hexagon is a geometric figure in which at least one vertex is pointing inward.

## Fundamental characteristics of a hexagon

Hexagons have the following fundamental characteristics:

- The sum of all its interior angles is 720°.
- Regular hexagons have all six sides with the same length.
- Regular hexagons have all six angles with the same measure.
- Each internal angle in a regular hexagon measures 120°.
- The total number of diagonals in a regular hexagon is 9.
- The sum of all exterior angles is 360 ° and each exterior angle measures 60°.

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## Important hexagon formulas

The following are the most important formulas for regular hexagons.

### Formula for the perimeter of a regular hexagon

A regular hexagon has all its sides of the same length, so its perimeter is given by:

$latex p=6s$ |

where *s* is the length of one of the sides of the hexagon.

### Formula for the area of a regular hexagon

The area of a regular hexagon is:

$latex A= \frac{3\sqrt{3}}{2}{{s}^2}$ |

where *s* is the length of one of the sides and *a* is the length of the apothem.

### Formula of the apothem of a regular hexagon

The apothem of a hexagon can be found using the following formulas:

$latex a= \frac{s}{2\tan(30°)}$ or $latex a=\frac{\sqrt{3}s}{2}$ |

where *s* is the length of one side of the hexagon.

## Examples of hexagon problems

### EXAMPLE 1

- A hexagon has sides of length 12 m. What is its perimeter?

**Solution:** We substitute $latex s=12$:

$latex p=6s$

$latex p=6(12)$

$latex p=72$

The perimeter is 72 m.

### EXAMPLE 2

- A hexagon has sides of length 10 m. What is its area?

**Solution:** We have $latex s=10$. Therefore, we use this value in the area formula:

$latex A= \frac{3\sqrt{3}}{2}{{s}^2}$

$latex A= \frac{3\sqrt{3}}{2}{{(10)}^2}$

$latex A=259.8$

The area of the hexagon is 259.8 m².

### EXAMPLE 3

- A hexagon has sides of length 8m. What is its apothem?

**Solution:** We use the apothem formula with $latex s=8$:

$latex a= \frac{\sqrt{3}s}{2}$

$latex a= \frac{\sqrt{3}(8)}{2}$

$latex a=6.93$

The length of the apothem is 6.93 m.

## Hexagon – Practice problems

## See also

Interested in learning more about hexagons? Take a look at these pages:

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