Geometry - Regular Polygon

• What Is a Polygon?
• Characteristics of Polygons
• Types of Polygons
• Definition of Regular Polygon
• Properties of Regular Polygon
• Comparison of Regular Polygons
• Solved Examples
• Practice Problems
• Frequently Asked Questions

What Is a Polygon?

A polygon is a two-dimensional shape identified by its sides and angles. It is a closed figure that is generated by the joining line segments. The term “polygon” is derived from the Greek words "poly,” meaning many, and "gonia,” meaning angle formation. Some images of polygons are shown below.

Characteristics of Polygons

1. Sides: The sides of the polygon are defined by the straight line segments, which cannot be curves or arcs.
2. Vertices: The line segments, after joining, form the corner points that are called vertices. A single corner is called the vertex.
3. Angles: The angles are formed by the intersection of line segments; the angles can be interior or exterior angles.

Types of Polygons 

1. Regular Polygons: These are the polygons that have all sides and angles equal. They are also symmetrical.
2. Irregular Polygons: Irregular polygons are polygons that do not have their sides and angles equal. These are not symmetrical.

Definition of Regular Polygon

A polygon is called a regular polygon if its sides and interior angles are congruent. For example, a square, an equilateral triangle, etc. So, the regular polygons are equilateral and equiangular.

Properties of Regular Polygon

The properties of regular polygons are:

`1`. Side lengths: All sides are of equal length. 

`2`. Interior angles: All its interior angles are equal. They can be found using the following formula.
`"Interior angles"="Sum of interior angles / Number of sides"`
 

`3`. Equal exterior angles: These are the angles formed by extending the sides of the polygons. The sum of its exterior angles is `360°`. They are also of equal measure.

`4`. The sum of interior angles: The sum of the interior angles can be calculated using the formula
`"Sum of interior angles" =(n-2)xx180°`
where `n=` number of sides 

`5`. Symmetry: Regular polygons are highly symmetrical, and if they are rotated, they will look exactly the same.

`6`. Order of rotational symmetry of a Regular Polygon: The order of rotational symmetry of a regular polygon is given by its number of sides, i.e., rotational symmetry`=n`. The angle of rotational symmetry is given as
`"Angle of rotational symmetry"= (360°)/n`

`7`. Diagonals: The diagonals are made by joining the non-adjacent vertices. The number of diagonals can be calculated as 

`"Number of diagonals"=(n(n-3))/2`
 

In the above image, the polygon has five sides. Therefore, `n=5`.
`"Number of diagonals"=(n(n-3))/2=(5xx(5-3))/2=5`

`8`. Perimeter: The perimeter can be calculated as the sum of all the sides. The perimeter can be given as,
`"Perimeter of a regular polygon"=nxx("Side length")`

`9`. Number of triangles: The number of triangles is formed by joining the diagonals at one vertex of the polygon.
`"Number of triangles"=n-2`

Comparison of Regular Polygons

Polygon	Number of Sides (n)	Interior Angle Measure	Exterior Angle Measure	The sum of the Interior Angles	Perimeter
Equilateral Triangle	`3`	`60°`	`120°`	`180°`	`3s`
Square	`4`	`90°`	`90°`	`360°`	`4s`
Pentagon	`5`	`108°`	`72°`	`540°`	`5s`
Hexagon	`6`	`120°`	`60°`	`720°`	`6s`
Heptagon	`7`	`128.57°`	`51.43°`	`900°`	`7s`
Octagon	`8`	`135°`	`45°`	`1080°`	`8s`
Nonagon	`9`	`140°`	`40°`	`1260°`	`9s`
Decagon	`10`	`144°`	`36°`	`1440°`	`10s`

Solved Examples

Example `1`: What is the measure of each interior angle of a regular heptagon?

Solution:

A regular heptagon has seven sides, i.e., `n=7`.

The sum of interior angles is calculated as, `(n-2)xx180°=(7-2)xx180°=900°`

Each interior angle can be calculated as, `(" sum of interior angles ")/(7)=(900^(@))/(7)=128.57^(@)`

Example `2`: Calculate the number of diagonals for a nonagon.

Solution: 

A nonagon has nine sides, i.e., `n=9`.

Number of diagonals`=(n(n-3))/2=(9xx(9-3))/2=(9xx6)/2=27`

A nonagon has `27` diagonals.

Example `3`: If an eleven-sided polygon has a side length of `14` cm, find its perimeter.

Solution: 

Here, side length `(s=14` `cm)`.

Perimeter of the polygon `=11s=11xx14=154` `cm`.

Example `4`: The interior angle of a regular polygon is `150°`. Find the number of sides.

Solution:

Here, the interior angle is `150°`.

The sum of interior angles `=(n-2)xx180°`

Each interior angle `=(" sum of interior angles ")/(n)=((n-2)xx180^(@))/(n)`

Therefore,

`150°=((n-2)xx180°)/n`

`150n=(n-2)xx180`

`150n=180n-360`

`360=180n-150n`

`360=30n`

`n=360/30`

Number of sides, `n=12`

Practice Problems

Q`1`. The measure of each interior angle of a regular polygon with `14` sides is ____.

1. `164°`
2. `154.28°`
3. `108.57°`
4. `164°`

Answer: b

Q`2`. The sum of all interior angles of a regular pentagon with each interior angle of `108°` is _____.

1. `640°`
2. `900°`
3. `540°`
4. `900°`

Answer: c

Q`3`. A polygon has `90` diagonals; find the number of sides of the polygon.

1. `10`
2. `12`
3. `15`
4. `18`

Answer: c

Q`4`. Can a regular polygon have an interior angle of `130°`?

1. Yes
2. No

Answer: b

Frequently Asked Questions

Q`1`. Can all polygons be inscribed in a circle?

Answer: No, all polygons cannot be inscribed in a circle. Most regular polygons can be inscribed in a circle, such as triangles, squares, pentagons, and hexagons, but polygons like regular heptagons cannot be inscribed in a circle.

Q`2`. In what real-life situations can regular polygons be used?

Answer: Regular polygons are very commonly used in fields such as architecture, design, and engineering for creating highly symmetrical shapes. 

Q`3`. Is a circle a regular polygon?

Answer: No, the circle is not a regular polygon, because a circle does not have sides, edges, and vertices and more importantly it does not have straight lines.