A polygon is a closed shape with three or more than three sides. The name of the polygon depends on the number of sides it has. For example, a polygon with four sides is named as quadrilateral. Now, as the name says, a hexagon has 6 sides. When all the sides of a polygon are of equal length, it is known as a regular polygon. Similarly, a regular hexagon is a closed geometrical shape with six equal sides.

## Properties of a Regular Hexagon

1) The first and main property that is derived from the name itself is that it has six sides and all are equal in length.

2) Since all sides are equal, therefore all angles in a regular hexagon are equal.

3) All exterior angles of a regular hexagon are of same measure which is equal to $60^{\circ}$. This can be evaluated from the fact that all exterior angles of a polygon add up to $360^{\circ}$ and one angle will thus be given by dividing it by number of sides.

4) Every interior angle of a regular hexagon is of measure $120^{\circ}$. This can be evaluated using the exterior angle measure and the straight line angle property.

5) Sum of the interior angles of a regular hexagon is $720^{\circ}$.

6) The triangles that are formed in the hexagon by joining all the vertices with the center of the hexagon are all equilateral and are equal in size.

7) There are nine diagonals in a regular hexagon and all are of equal length.

8) The apothem (Ap) of a regular hexagon can be calculated using the formula below, if ‘$l$’ which is the length of the line joining the vertex and the center of the hexagon are known.

$Ap$ = $\sqrt{(l^{2} – (\frac{l}{2})^{2})}$ = $(\sqrt{3})\ \times$ $\frac{l}{2}$

## Perimeter of a Regular Hexagon

The perimeter of any shape is the sum of the lengths of all its sides. In a regular hexagon we have $6$ sides all equal, say, ‘$s$’.

$Perimeter\ of\ regular\ hexagon$ = $6\ \times\ s$

That is, perimeter of a regular hexagon can be determined by finding the product of the number of sides with $6$.

## Area of Regular Hexagon

The area of a regular hexagon can be evaluated if the side and the apothem is known or determined already.

Area of regular hexagon = $\frac{(perimeter\ \times\ apothem)}{2}$

Area of a regular hexagon = Sum of areas of six equilateral triangles making up the hexagon.

Area of equilateral triangle = $\sqrt{3}\ \times$ $\frac{(Length\ of\ each\ side\ of\ regular\ hexagon)^{2}}{4}$

Area of each triangle = $\frac{1}{2}$ $\times$ side of hexagon $\times$ perpendicular distance from the center to the side of the hexagon

Area of each triangle = $\frac{1}{2}$ $\times$ side of hexagon $\times$ apothem

Apothem is the length of the segment that joins the center to the midpoint of the hexagon and also perpendicular to the side.

Area of regular hexagon = $\frac{1}{2}$ $\times$ Perimeter of the regular hexagon $\times$ apothem

Perimeter of regular hexagon = Sum of the six sides of the regular hexagon

Therefore, Area of regular hexagon = $\frac{1}{2}$ $\times\ 6\ \times$ Length of each side $\times$ apothem

= $3\ \times$ Length of each side $\times$ apothem

## Regular Hexagon Angles

There are two types of angles in a polygon: exterior angle and interior angle. The interior angle is the one that is made by two adjacent sides of the polygon. The exterior is made by a side and another produced side of the polygon.

In any polygon, the sum of all exterior angles is $360^{\circ}$. We have six equal sides in a hexagon which implies six equal exterior angles and therefore each exterior angle of a regular hexagon is of $60^{\circ}$.

Now, we know that the interior and exterior angle in a polygon form on a straight line. Hence, the interior angle can be evaluated by finding the supplement of the exterior angle which is in this case $120^{\circ}$. So, all six interior angles in a regular hexagon measure $120^{\circ}$.

The total sum of all the angles of a regular hexagon is given by: $120\ \times\ 6$ = $720^{\circ}$

## Constructing Regular Hexagon

If we are not given any measurements and we need to draw a regular hexagon, then we follow the following steps.

1) Draw a circle and mark any point on it.

2) With same radius of compass as that of the circle, start making arcs intersecting the circle starting from the point we marked and then taking the point of intersection obtained as new point to make another arc. Continue this till we have six such points.

3) Join all points in adjacent manner and obtain the required hexagon.

If we are given a side of a regular hexagon, then we follow the steps below to construct the required hexagon.

1) Draw a line of the given length.

2) With radius more than half of this line, make two arcs from the two end points of the line segment intersecting each other at a point. This point is the center of the hexagon.

3) Now, with the center of the hexagon make a circle with same radius as in step 2.

4) With the first point of the line segment draw another arc with same radius intersecting the circle. With this new point, draw next arc as this one and keep on continuing till we reach the second end point of the line segment. We will have six vertices in total now.