Regular Polygon

What is a Regular polygon?

Examples of regular polygons are equilateral triangles, squares, rhombuses, and so on.

A polygon will also have diagonals of the same length. Regular polygons are mostly convex by nature. On the other hand, concave regular polygons are sometimes star-shaped. We will be discussing the properties of regular convex polygons in detail.

Regular polygon properties

Circumcircle and Incircle

There are two important circles that can be drawn on a regular polygon.

• The circumcircle lies outside the convex regular polygon and passes through all its vertices. The radius of the circumcircle is the distance from the center of the polygon to any of its vertex.

• The incircle passes through the mid-point of all the sides of the polygon and lies inside the regular polygon. The radius of the incircle is the distance between the center and a midpoint of any side. This distance is also called the apothem of the polygon.

These properties of an incircle, circumcircle, and apothem can only be found in regular polygons.

Circumcircle, incircle and apothem, mathisfun.com

So what can we do using these properties?. One interesting application is being able to calculate the area of a regular polygon using the apothem. Any regular polygon can be broken down into triangles, Combining this with the apothem we can estimate the areas of any regular polygon of side N.

Regular polygon examples

Regular polygons with 3 sides are called equilateral triangles, 4 sides are called squares. Regular polygons with more than four sides are denoted with a 'regular' preceding the name of the polygon. For example, a pentagon with equal sides and angles is called a regular pentagon. Below are a few examples of a regular (equiangular) convex polygon.

Regular polygon examples, mathworld.wolfram.com

Formulas for regular polygons

Regular polygons have a few interesting properties associated with each of their attributes. We look at them in the following sections.

Exterior angles

At any vertex of a polygon, there are 2 angles the interior and exterior. The exterior angle is obtained by the angle between an extended edge and its consecutive edge.

Exterior angles of a regular polygon, geogebra.org

In a regular convex polygon, the sum of all exterior angles is always 360° It can also be written as,

$\varnothing =360/N,whereNisthenumberofsidesand\varnothing istheexteriorangle.$

Interior Angles

The interior angles are formed between two adjacent sides of a polygon. The sum of interior angles of a polygon will depend on the number of sides that it has. For example, all triangles will have a total sum of 180°, quadrilaterals will have a sum of 360°, and so on. But what about a polygon with hundred sides.

Interior angle of a regular polygon, socratic.org

The sum of the interior and exterior angles at a vertex is always equal to 180°. Using this relation we can derive a general equation that can be used to find the interior angles of any polygon by having the number of sides.

$$\begin{gathered} InteriorAngle=180°-ExteriorAngle \\ WealreadyknowtheExteriorangle=\frac{360°}{N},so \\ InteriorAngle=180°-\frac{360°}{N} \\ InteriorAngle=180°-\frac{360°}{N} \\ =(N\times \frac{180°}{N})-(2\times \frac{180°}{N}) \\ =(N-2)\times \frac{180°}{N} \\ \mathit{T}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathit{f}\mathit{o}\mathit{r}\mathit{e}\mathbf{,}\mathbf{}\mathit{I}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{i}\mathit{o}\mathit{r}\mathbf{}\mathit{a}\mathit{n}\mathit{g}\mathit{l}\mathit{e}\mathbf{=}\mathbf{}\mathbf{(}\mathit{N}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{}\mathbf{\times }\mathbf{}\frac{\mathbf{180}\mathbf{°}}{\mathbf{N}} \\ \mathit{A}\mathit{n}\mathit{d}\mathbf{}\mathit{S}\mathit{u}\mathit{m}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{a}\mathit{l}\mathit{l}\mathbf{}\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{i}\mathit{o}\mathit{r}\mathbf{}\mathit{a}\mathit{n}\mathit{l}\mathit{g}\mathit{l}\mathit{e}\mathit{s}\mathbf{=}\mathbf{(}\mathit{N}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{180}\mathbf{°} \end{gathered}$$

Diagonals of a Convex polygon

In a polygon with more than 3 sides, a diagonal is a line segment between any two non-consecutive points. Unlike the concave polygons, the diagonals of a convex polygon will always lie inside the figure. If a polygon has 'N' sides, then the number of diagonals is equal to:

$Numberofdiagonals=\frac{N(N-3)}{2}$.

This brings us to the end of this article. Let us refresh what we've learned so far.