Area of Regular Polygons

What are regular polygons?

A regular polygon is a type of polygon in which all sides are equal to each other and all of the angles are equal as well. Also, the measure of all the interior and exterior angles are equal, respectively.

Regular polygons include equilateral triangles (3 sides), squares (4 sides), regular pentagons (5 sides), regular hexagons (6 sides), etc.

Regular polygons, StudySmarter Originals

Note that if the polygon is not a regular polygon (that is, it does not have equal side lengths and equal angles), then it can be called an irregular polygon. For example, a rectangle or a quadrilateral can be called an irregular polygon.

Properties and elements of a regular polygon

Let us first consider the properties and elements of a regular polygon before beginning the discussion on its area.

Any regular polygon has different parts like a radius, apothem, side, incircle, circumcircle, and center. Let's discuss the concept of the apothem.

Apothem of the regular polygon, StudySmarter Originals

The apothem is the line from the center to one side that is perpendicular to that side and is denoted by the letter a.

To find the apothem of the polygon, we first need to find its center. For a polygon with an even number of sides, this can be done by drawing at least two lines between opposing corners and seeing where they intersect. The intersection will be the center. If the polygon has an odd number of sides, you will need to draw lines between one corner and the midpoint of the opposing side instead.

Diagonals and center of regular polygon, StudySmarter Originals

The properties of a regular polygon include:

• All sides of a regular polygon are equal.
• All interior and exterior angles are equal respectively.
• Each angle of a regular polygon is equal to $\frac{\left(n-2\right)\times 180°}{n}.$
• The regular polygon exists for 3 or more sides.

Formula for the area of regular polygons

Now you know everything you need in order to use the formula for finding the area of a regular polygon. The formula for the area of a regular polygon is:

$Area=\frac{a\times p}{2}$

where a is the apothem and p is the perimeter. The perimeter of a regular polygon can be found by multiplying the length of one side by the total number of sides.

Derivation of area formula using a right triangle

Let's take a look at this formula's derivation in order to understand where it comes from. We can derive the formula for the area of regular polygons by using a right triangle to construct n triangles of equal size within a polygon of n sides. Then, we can add all the areas of the individual triangles together to find the area of the whole polygon. For example, a square has four sides, so can therefore be divided into four triangles as shown below.

Division of square into four equal parts, StudySmarter Originals

Here, x is the length of one side and a is the apothem. Now, you might remember that the area of a triangle is equal to $\frac{b\times h}{2}$, where b is the base of the triangle and h is the height.

In this case,

so, the area for one triangle inside the square can be expressed as:

$\frac{a\times x}{2}$

Because there are four triangles, we need to multiply this by four to get the total area of the square. This gives:

$\Rightarrow 4\times \frac{a\times x}{2}=\frac{a\times 4x}{2}$

Consider the term, 4x. You may have already noticed that the perimeter of the square is the sum of its four sides, equal to $4x$. So, we can substitute$p=4x$ back into our equation to get the general formula of the area of a regular polygon:

$Area=\frac{a\times p}{2}$

Finding the area of regular polygons using trigonometry

The length of the apothem or perimeter might not always be given in a question about regular polygons. However, in such cases, we can use our knowledge of trigonometry to determine the missing information if we know the side length and the angle size. Let's consider how trigonometry relates to regular polygons with the following example scenario.

We are given a regular polygon with n sides, with radius r and side length x.

Regular polygon with n(=5) sides, StudySmarter Originals

We know that angle $\theta$ will be $\frac{360°}{n}.$ Let's consider one section of the polygon, as shown in the figure below. In this section, we draw an apothem from the center, splitting it into two right triangles.

One part of the regular polygon, StudySmarter Originals

We know that $\angle BAC$ is $\theta ,$ then $\angle BAD\&\angle DAC$ will be $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, respectively, as the apothem is the perpendicular bisector from the center. Now, by calculating the area of any one of the right triangles, we can find the area of the regular polygon. Hence, the area of the right triangle is:

$Area=\frac{1}{2}\times a\times \left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$

where, $a=r\cos \left(\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right),\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=r\sin \left(\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right).$

The area of the polygon section is twice the area of the right triangle.

$$\begin{gathered} \Rightarrow Areaofonepartofpolygon=2\times areaofrighttriangle \\ =a\times \left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) \end{gathered}$$

Now, considering all sections of the polygon, the whole polygon's area is n times the area of one section.

$$\begin{gathered} \Rightarrow \mathit{A}\mathit{r}\mathit{e}\mathit{a}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{r}\mathit{e}\mathit{g}\mathit{u}\mathit{l}\mathit{a}\mathit{r}\mathbf{}\mathit{p}\mathit{o}\mathit{l}\mathit{y}\mathit{g}\mathit{o}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathbf{\times }\mathit{a}\mathit{r}\mathit{e}\mathit{a}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{o}\mathit{n}\mathit{e}\mathbf{}\mathit{p}\mathit{a}\mathit{r}\mathit{t}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{p}\mathit{o}\mathit{l}\mathit{y}\mathit{g}\mathit{o}\mathit{n} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathbf{\times }\left(a\times \left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\right) \end{gathered}$$

Area of regular polygons examples and problems

Let us see some solved examples and problems dealing with the area of regular polygons.