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# Regular Polygon

A polygon can be defined as a closed shape with straight edges. Since they need to be closed, they'll always have three or more sides and having straight edges removes circles and ovals from this category. Polygons can be classified on a lot of different criteria, from the number of sides to their interior and exterior angles.

The criteria that determine whether a shape is convex or concave is the magnitude of interior angles.

If all the interior angles are less than 180° each, then the shape is classified as convex. Whereas if any one of the interior angles is greater than 180°, then the shape is concave. Convex polygons are further classified into regular or irregular polygons depending on the length of the sides and interior angles.

In this article, we go through what a regular polygon is, its properties, and a few examples.

## What is a Regular polygon?

A regular polygon has sides of equal length and equal interior angles.

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.

Calculate the area of a hexagon with side s and apothem I.

Area of a regular polygon, mathisfun.com

Solution

Divide the hexagon into six triangles as shown in the image below. We observe the following

• The base of the triangle is equal to the side of the polygon (s).
• The height of the triangle is nothing but the apothem of the polygon (l).

To get the area of a polygon all we need is to calculate the area of one triangle and multiply it by the number of sides.

Therefore, the Area of the hexagon = $6×\frac{1}{2}Xs×l$

## 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.

$InteriorAngle=180°-ExteriorAngle\phantom{\rule{0ex}{0ex}}WealreadyknowtheExteriorangle=\frac{360°}{N},so\phantom{\rule{0ex}{0ex}}InteriorAngle=180°-\frac{360°}{N}\phantom{\rule{0ex}{0ex}}InteriorAngle=180°-\frac{360°}{N}\phantom{\rule{0ex}{0ex}}=\left(N×\frac{180°}{N}\right)-\left(2×\frac{180°}{N}\right)\phantom{\rule{0ex}{0ex}}=\left(N-2\right)×\frac{180°}{N}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbit{T}\mathbit{h}\mathbit{e}\mathbit{r}\mathbit{e}\mathbit{f}\mathbit{o}\mathbit{r}\mathbit{e}\mathbf{,}\mathbf{}\mathbit{I}\mathbit{n}\mathbit{t}\mathbit{e}\mathbit{r}\mathbit{i}\mathbit{o}\mathbit{r}\mathbf{}\mathbit{a}\mathbit{n}\mathbit{g}\mathbit{l}\mathbit{e}\mathbf{=}\mathbf{}\mathbf{\left(}\mathbit{N}\mathbf{-}\mathbf{2}\mathbf{\right)}\mathbf{}\mathbf{×}\mathbf{}\frac{\mathbf{180}\mathbf{°}}{\mathbf{N}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbit{A}\mathbit{n}\mathbit{d}\mathbf{}\mathbit{S}\mathbit{u}\mathbit{m}\mathbf{}\mathbit{o}\mathbit{f}\mathbf{}\mathbit{a}\mathbit{l}\mathbit{l}\mathbf{}\mathbit{i}\mathbit{n}\mathbit{t}\mathbit{e}\mathbit{r}\mathbit{i}\mathbit{o}\mathbit{r}\mathbf{}\mathbit{a}\mathbit{n}\mathbit{l}\mathbit{g}\mathbit{l}\mathbit{e}\mathbit{s}\mathbf{=}\mathbf{\left(}\mathbit{N}\mathbf{-}\mathbf{2}\mathbf{\right)}\mathbf{}\mathbf{×}\mathbf{}\mathbf{180}\mathbf{°}$

Calculate the exterior and interior angles and the sum of all interior angles for a regular decagon.

Angles of a regular decagon, tutors.com

Solution

We know that, Exterior angle = $\frac{360°}{N}=\frac{360°}{10}=36°$

Similarly, Interior angle = $180°-exterioirangle=180°-36°=144°$

Therefore sum of all interior angles = N X 144° = 10 X 144° = 1440°

### 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:

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

Calculate the number of diagonals in a heptagon.

Solution

Applying the formula we get

$Numberofdiagonals=\frac{N\left(N-3\right)}{2}=\frac{7\left(7-3\right)}{2}\phantom{\rule{0ex}{0ex}}=\mathbf{14}\mathbf{}\mathbit{d}\mathbit{i}\mathbit{a}\mathbit{g}\mathbit{o}\mathbit{n}\mathbit{a}\mathbit{l}\mathbf{s}$

Diagonals of a regular pentagon, brainly.in

We get a total of 14 diagonals, which are shown in the figure above.

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

## Regular Polygons - Key takeaways

• A polygon with equal sides and interior angles is called a regular polygon.
• All the diagonals of a regular polygon are equal in length.
• The circumcircle passes through all the vertices of a regular polygon. The radius of the circumcircle is called the circumradius.
• The Incircle passes through the midpoints of each side. The radius of the incircle is called the apothem of the polygon.
• Every regular polygon can be broken down into smaller triangles which can be used to calculate their area.
• All the vertices of a regular polygon are equidistant from its center.
• The sum of exterior angles in a regular polygon is always equal to 360°.
• The sum of all interior angles for a regular polygon is given by $\mathbf{\left(}\mathbit{N}\mathbf{-}\mathbf{2}\mathbf{\right)}\mathbf{}\mathbf{×}\mathbf{}\mathbf{180}\mathbf{°}$
• The number of diagonals for a polygon with 'N' sides is given by$\frac{\mathbf{N}\mathbf{\left(}\mathbf{N}\mathbf{-}\mathbf{3}\mathbf{\right)}}{\mathbf{2}}$

A polygon with equal sides and interior angles is called a regular polygon.

The minimum number of sides for a regular polygon is 3 ( Equilateral triangle), it has no upper limit.

Equilateral triangles, squares, rhombuses are examples of regular polygons.

The area of a regular polygon can be calculated by dividing it into triangles. To get the area of the whole polygon, just add up the areas of all the little triangles.

Regular polygons will have different shapes depending on the number of sides.

## Final Regular Polygon Quiz

Question

What is a regular polygon?

A polygon with equal sides and interior angles is called a regular polygon.

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Question

How many sides does a regular polygon have?

The minimum number of sides for a regular polygon is 3 ( Equilateral triangle), it has no upper limit.

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Question

What is an example of a regular polygon?

Equilateral triangles, squares, rhombuses are examples of regular polygons.

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Question

How to find the area of a regular polygon?

The area of a regular polygon can be calculated by dividing it into triangles. To get the area of the whole polygon, just add up the areas of all the little triangles.

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Question

What is a circumcircle of a 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.

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Question

What is the incircle of a polygon?

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.

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Question

A polygon with equal side lengths and angle measures.

Regular polygon

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Question

What is the sum of all exterior angles in a regular polygon?

360°

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Question

What is the apothem of a 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.

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