In our surroundings, we observe numerous forms, from tables and clocks to sandwiches and pizzas. In the study of geometry, we explore various forms like triangles and squares, all of which are instances of polygons. A polygon is defined as a two-dimensional enclosed figure crafted from straight lines.

This article aims to delve into the understanding of areas for regular polygons through the calculation of the apothem.

## Understanding Regular Polygons

A regular polygon stands as a shape in which each side and angle are of equal length and measure respectively, ensuring uniformity in its interior and exterior angles.

Geometric shapes named regular polygons are recognized by their equilateral (all sides of equal length) and equiangular (all angles of equal measure) properties.

Regular polygons are identified as equilateral triangles, squares, pentagons, hexagons, and more, sharing equilateral and equiangular characteristics.

Should a polygon display unequal side lengths and angles, it is termed an irregular polygon, with rectangles and quadrilaterals serving as examples.

### Attributes and Components of a Regular Polygon

Prior to diving into the calculation of a regular polygon's area, it is essential to outline its attributes and components.

Emphasizing congruence, regular polygons possess identical sides and angles alongside components such as the radius, apothem, side, incircle, circumcircle, and center. The focal point of our discussion will be the apothem concept.

The** apothem** within a polygon is illustrated as a line stretching from the polygon's center to the midpoint of its side, maintaining a perpendicular alignment to the side in question.

Denoted by the symbol a, the apothem signifies a perpendicular connection from the polygon’s center to its side.

To ascertain a polygon’s apothem, identifying its center is paramount. In polygons with an even side count, connecting opposing vertices and locating their intersection reveals the center. Conversely, in polygons with an odd side count, connections between a vertex and the opposing side's midpoint are necessary.

Key properties of a regular polygon encompass:

- Uniform side lengths across the polygon.
- Equal measure of all interior and exterior angles.
- Each polygon angle computes to $\frac{\left(n-2\right)\times 180\xb0}{n}.$
- The existence of a regular polygon is validated with 3 or more sides.

## Area Calculation Formula for Regular Polygons

Equipped with the necessary knowledge, we can now apply the formula to compute a regular polygon's area, expressed as:

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

where a represents the apothem and p the perimeter. The **perimeter of a regular polygon** is obtained by multiplying a single side's length by the count of sides.

### Area Formula Derivation through Right Triangle Construction

To comprehend the formula's origin, let's explore its derivation by segmenting a polygon into n identical right-angled triangles, then combining their areas to deduce the polygon's total area. For instance, a square, with four sides, can thus be segmented into four right-angled triangles.

The computation of a regular polygon's area, like that of a square with sides x and apothem a, follows the formula:

Area = (a * p) / 2

with p representing the polygon's perimeter, which equals 4x for a square.

Upon substituting the respective values, we arrive at:

Area = (a * 4x) / 2 = 2ax

Conclusively, the calculation for a regular polygon's area equals twice the multiplication of the apothem by the side length.

## Employing Trigonometry for Regular Polygon Area Calculation

In instances where a regular polygon's apothem or perimeter is undisclosed, trigonometry offers a solution by determining the missing elements given known side lengths and angle measures. Let's investigate this application through a scenario involving a regular polygon of n sides, radius r, and side length x.

Regular polygon with n(=5) sides

The internal angle $\theta $ equates to $\frac{360\xb0}{n},$, enabling us to isolate a polygon segment by extending an apothem from the center, thus forming two symmetric right triangles.

With $\angle BAC$ as $\theta ,$, $\angle BAD,\angle DAC$ equal $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ as the apothem bisects perpendicularly. Determining one right triangle's area facilitates the entire polygon's area calculation.

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

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

Thus, the polygon segment's area doubles that of the right triangle.

$\Rightarrow Areaofonesection=2\times areaofrighttriangle$

## Illustrative Examples and Problems on Regular Polygon Area

Below are illustrative scenarios and exercises pertaining to the computation of areas within regular polygons.

Deduce the area for the stipulated regular polygon.

**Resolution:** With $a=14,side=28\sqrt{3}$ as provided measurements, the circumference p is calculated as:

$p=3\times sde=3\times 28\sqrt{3}=145.5$

Accordingly, the area of the regular polygon is deduced to be:

$Area=\frac{a\times p}{2}\phantom{\rule{0ex}{0ex}}=\frac{14\times 145.5}{2}\phantom{\rule{0ex}{0ex}}=1018.5$

Compute the area for a hexagon when given a side length of 4 cm and an apothem of 3.46 cm.

**Resolution:** With absence of apothem specifics, the hexagon's boundary length becomes crucial for the area formula application.

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

The boundary is determined by multiplying the dimension of one side by the total number of sides.

$\Rightarrow p=4\times 6=24cm$

Inserting the deduced values into the area equation yields:

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

Estimate the area of a square yard with each side extending 3 feet. How should one proceed?

**Resolution:** Addressing a square polygon with a dimension $x=3ft.$, we seek the apothem for the area's determination.

Segmenting the square yields four equitable parts. The angular measure of one polygon portion (from the center’s perspective) computes as $\theta =\frac{360\xb0}{n}=\frac{360\xb0}{4}=90\xb0$. Further division into two equal right angles renders $\frac{\theta}{2}=\frac{90\xb0}{2}=45\xb0.$

Employing trigonometry leads to apothem a determination as:

$\mathrm{tan}\left(\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)=\frac{oppside}{adjside}\phantom{\rule{0ex}{0ex}}\mathrm{tan}45\xb0=\frac{\left(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}{a}\phantom{\rule{0ex}{0ex}}\Rightarrow a=\frac{\left(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}{\mathrm{tan}45\xb0}\phantom{\rule{0ex}{0ex}}=\frac{\left(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}{1}\phantom{\rule{0ex}{0ex}}=1.5$

With these values, the regular polygon's area can be computed, resulting in:

$Area=n\times \left(a\times \left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\right)\phantom{\rule{0ex}{0ex}}=4\times \left(1.5\times 1.5\right)\phantom{\rule{0ex}{0ex}}=9f{t}^{2}$