A regular polygon is a polygon whose sides (or edges) and angles are all the same. How can we remember this? The word regular denotes something common, without changes or excesses, therefore, you can remember that in this polygon there are no changes, meaning all its sides are equal and all its angles are also equal.

Examples of regular polygons:

Equilateral triangle - All its sides are equal and all its angles are also equal.

Square - All its sides are equal and all its angles measure $90$.

Regular pentagon - It has $5$ equal sides and $5$ equal angles.

Regular polygon

Regular tessellation

Regular tessellation is created by covering a surface with isometric (identical) copies of an initial figure and "attaching" them to the edge of another regular polygon. This way, it happens that the sum of the angles at each vertex is $360$! We can achieve $3$ tessellations of this type with triangles, squares, and hexagons. Let's see it in an illustration:

Tessellation with triangles

Tessellation with squares

Tessellation with regular hexagons

Sum of the angles of a regular polygon

In fact, the formula to find the sum of the angles of a regular polygon is the same as the formula to find the sum of the angles of any polygon, which is:

By observing this formula, you have surely realized that the sum of the angles of a polygon depends on the number of edges the polygon has.

Let's practice: Let's look at the following polygon.

Find the size of angle $H$ and determine if it is a regular polygon knowing that:

All sides are equal

$∢A=∢B=∢C=∢D=∢E=∢F=∢G=135$

Solution: To determine if the polygon is regular, we must show that all its sides and all its angles are equal. We know that all its sides are equal. To show that all its angles are equal, we will calculate the sum of the angles using the following formula: $180\times(8-2)=1080$ $1080$ is the total of the angles of the polygon. We know that each of the $7$ angles measures $135$ so we will solve for the measure of the unknown angle: $1080-(7\times135)=$ $1080-945=135$ We have discovered that angle $H$ also equals $135$, therefore, the polygon is regular since all its sides and all its angles are equal.

We already know that in a regular polygon all the angles are equal. Therefore, if you look at the formula, you will see that it takes the sum of the angles of the polygon and divides it by the number of angles it has, since, as we know, they all measure the same.

Let's practice: Given a regular polygon with $7$ sides. How much does each angle measure?

Solution: We simply place in the formula $n=7$ and we get:

$\frac{180\times(7-2)}7=$

$\frac{180\times5}7=128.571$ Each angle of the regular polygon is equivalent to $128.571$

Another exercise: Given a regular octagon: Find the measure of its angle.

Solution: We simply place in the formula $n=8$ and we get:

$\frac{180\times(8-2)}8=$

$\frac{180\times6}8=135$

Each angle of the regular octagon is equivalent to $135$

To calculate the area of a regular hexagon, all you need to do is remember the following formula:

$6 \times \frac{a^2 \sqrt3}{4}$

where, $a$ = length of the side of the regular hexagon

Let's practice! Given the following regular hexagon We know that the length of one side of the hexagon is $5$ cm. Find the area of the regular hexagon.

Solution: We will proceed according to the formula and set $a=5$. We will obtain: $6\times\frac{5^2 \sqrt3}{4}=64.95$

The area of the regular hexagon is $64.95$ cm^{2}.