# Area of Regular Polygons

Everything around us has a particular shape, whether it's the table, clock, or food items like sandwiches or pizza. Especially in geometry, we have seen and studied different shapes like triangles or squares and many more. These shapes are some examples of polygons. Recall that a polygon is a two-dimensional closed shape formed using straight lines.

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In this article, we will understand the concept of the area of regular polygons, by finding the apothem.

## 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 are geometric figures where all sides have the same length (equilateral) and all angles have the same size (equiangular).

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.

The apothem of a polygon is a segment going from the center of the polygon to the midpoint of one of the sides. This means that it is perpendicular to one of the sides of the polygon.

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 $\frac{\left(n-2\right)×180°}{n}.$
• 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:

$Area=\frac{a×p}{2}$

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 $\frac{b×h}{2}$, where b is the base of the triangle and h is the height.

In this case,

$b=x$ and $h=a$,

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

$\frac{a×x}{2}$

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

$⇒4×\frac{a×x}{2}=\frac{a×4x}{2}$

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:

$Area=\frac{a×p}{2}$

## 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 $\theta }{2}$, 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:

$Area=\frac{1}{2}×a×\left(x}{2}\right)$

where, $a=r\mathrm{cos}\left(\theta }{2}\right),x}{2}=r\mathrm{sin}\left(\theta }{2}\right).$

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

$⇒Areaofonepartofpolygon=2×areaofrighttriangle\phantom{\rule{0ex}{0ex}}=a×\left(x}{2}\right)$

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

$⇒\mathbit{A}\mathbit{r}\mathbit{e}\mathbit{a}\mathbf{}\mathbit{o}\mathbit{f}\mathbf{}\mathbit{r}\mathbit{e}\mathbit{g}\mathbit{u}\mathbit{l}\mathbit{a}\mathbit{r}\mathbf{}\mathbit{p}\mathbit{o}\mathbit{l}\mathbit{y}\mathbit{g}\mathbit{o}\mathbit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbit{n}\mathbf{×}\mathbit{a}\mathbit{r}\mathbit{e}\mathbit{a}\mathbf{}\mathbit{o}\mathbit{f}\mathbf{}\mathbit{o}\mathbit{n}\mathbit{e}\mathbf{}\mathbit{p}\mathbit{a}\mathbit{r}\mathbit{t}\mathbf{}\mathbit{o}\mathbit{f}\mathbf{}\mathbit{p}\mathbit{o}\mathbit{l}\mathbit{y}\mathbit{g}\mathbit{o}\mathbit{n}\phantom{\rule{0ex}{0ex}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbit{n}\mathbf{×}\left(a×\left(x}{2}\right)\right)$

## Area of regular polygons examples and problems

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

Find the area of the given regular polygon.

Regular polygon, StudySmarter Originals

Solution: Here we are given that $a=14,side=28\sqrt{3}$. So, perimeter p is:

$p=3×side=3×28\sqrt{3}=145.5$

Hence, the area of the regular polygon is:

id="2951752" role="math" $Area=\frac{a×p}{2}\phantom{\rule{0ex}{0ex}}=\frac{14×145.5}{2}\phantom{\rule{0ex}{0ex}}=1018.5$

Find the area of a hexagon with a side length of 4 cm and an apothem of 3.46 cm.

Solution: As the apothem is already given in the question, we only need to find the perimeter of the hexagon to use the area formula.

$Area=\frac{a×p}{2}$

The perimeter is the length of one side multiplied by the number of sides.

$⇒p=4×6=24cm$

Now substituting all the values in the formula of area, we get:

$Area=\frac{24×3.46}{2}=41.52c{m}^{2}$

Suppose a square yard has a length of 3 feet. What is the area of this yard?

Solution: We are given a square polygon with a length $x=3ft.$ We need to calculate the value of the apothem to find the area.

Square polygon with side 3 ft., StudySmarter Originals

First, let's divide the square into four equal sections. The angle of one section of the polygon (with respect to the center) is $\theta =\frac{360°}{n}=\frac{360°}{4}=90°$. As each section can be segmented into two right triangles, the angle associated with one right triangle is $\frac{\theta }{2}=\frac{90°}{2}=45°.$

Now, we can use a trigonometric ratio to evaluate the right triangle. We can find the value of the apothem a as:

Now, by substituting all values into the formula, we calculate the area of the regular polygon:

$Area=n×\left(a×\left(x}{2}\right)\right)\phantom{\rule{0ex}{0ex}}=4×\left(1.5×1.5\right)\phantom{\rule{0ex}{0ex}}=9f{t}^{2}$

So, the area of the yard is 9 square feet.

## Area of regular polygons - Key takeaways

• A regular polygon is equilateral and equiangular.
• The apothem of a polygon is a segment going from the center of the polygon to the midpoint of one of the sides.
• The perimeter of a regular polygon can be found by multiplying the length of one side by the number of sides.
• The formula for finding the area of a regular polygon is $Area=\frac{a×p}{2}$.
• The apothem may be worked out geometrically using trigonometry.

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How to find the area of a regular polygon?

The area of a regular polygon can be found using the formula area =(ap)/2 where a is the apothem and p is the perimeter

What kinds of regular polygons are symmetrical?

All regular polygons are symmetrical. the number of axes of symmetry is equal to the number of sides.

What are the properties of a regular polygon?

A regular polygon is equilateral (equal side lengths) and equiangular (equal angle sizes)

What is the formula for finding the area of a regular polygon

The formula to find the area of regular polygon is:

Area=(a*p)/2

How to find regular polygon using trigonometry?

The area of regular polygon is calculated with the help of right triangle and trigonometric ratio.

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