A polygon is a two-dimensional closed shape. It is having many sides. Each side will be a straight line segment. In geometry, a hexagon is also a type of polygon with 6 sides. If the hexagon is regular, then the lengths of all the sides will be the same as well as all the angles will be equal. Clearly sides of a regular hexagon are congruent. This article will discuss the hexagon and hexagon formula to find out the measurements of it. Also, some suitable examples will help students to understand it. Let us learn the concept!

**What is Regular Hexagon?**

A regular hexagon is a kind of polygon with 6 equal sides. Its properties are:

- It is having six sides and six angles.
- Lengths of all the sides are equal.
- Measurements of all the angles are equal.
- The total number of diagonals in it is 9.
- The sum of all interior angles will be 720 degrees. Also, each interior angle will be 120 degrees.
- The sum of all exterior angles will be 360 degrees. Also, each exterior angle will be 60 degrees.
- The area of a hexagon will be the region occupied inside its boundary.
- The perimeter of it will be the total length of all 6 sides of it.
- The shape has nine diagonals, means the lines between the interior angles.

Source: en.wikipedia.org

**Formulas for Hexagon**

### 1] Area of a Hexagon

To find out the area of a hexagon, we have to divide it into small six isosceles triangles. Then, we will find the area of one of the triangles and finally, we will multiply by 6 to find the total area of the polygon.

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

### 2] Permiter of a Hexagon

Perimeter will be the sum of all 6 side lengths. The formula for the perimeter of a hexagon is given as:

\(P = 6 \times a\)

### 3] Diagonal of a Hexagon

There is no standard formula to find out the diagonals of irregular hexagons. In regular hexagons, there are nine diagonals in six equilateral triangles. So, it is easy to determine the length of each diagonal line. If one side of the hexagon is given then it is easy to calculate the all diagonal. It is having diagonals of two different lengths.

Short diagonal : \(d_{1} = \sqrt{3} \times a\)

Long Diagonal: \(d_{2} = 2 \times a\)

a | Length of one side |

A | Area of hexagon |

P | Perimeter of Hexagon |

\(d_{1}\) | Short diagonal |

\(d_{2}\) | Long diagonal |

**Solved Examples for ****Hexagon Formula**

Q.1: Calculate the area and perimeter of a given regular hexagon whose side is 4 cm.

Solution: Given parameters are,

Side of the hexagon, a = 4 cm

Thus area of Hexagon will be,

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

\(=\frac{3\sqrt{3}}{ 2} \times 4^{2}\\\)

\(= 24\sqrt{3} sqcm.\)

The perimeter of the hexagon will be,

\(P = 6 \times a = 6 \times 4\)

= 24 cm

So, the area of the hexagon will be \(24\sqrt{3} sqcm.\), and perimeter will be 24

Q.2: Perimeter of a regular hexagonal board is 48 cm. Compute the area of this board.

Solution: Given parameters are:

Perimeter, P = 48 cm

Also, formula for perimeter is,

\(P = 6 \times a\)

Thus side,\( a = \frac {P}{6} = \frac {48}{6}\)

a = 8 cm

Now, Area of an Hexagon formula is:

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

\(= \frac{3\sqrt{3}}{ 2} \times 8^{2}\\\)

\(= 96 \sqrt{3} sqcm\)

So, area of hexagon will be \(96 \sqrt{3} \;sq\;cm\)