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# Nonagon: Definition, Interior Angles and Properties

## Introduction of Nonagon

Last updated date: 27th Mar 2023
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A polygon is a plane figure in geometry that is made from a finite number of straight-line segments. Different types of polygons are given in the following image, for example, a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon.

A nonagon is a polygon that contains a total of nine sides. It is a two-dimensional figure designed from straight line segments. It is a closed figure having nine sides, 27 diagonals, and nine vertices.

We will learn about interior angles, types, shapes, and sides of nonagon in detail in the article.

## Definition of Nonagon

When a polygon has a total of nine sides then it is called a nonagon. The other names of nonagon are Enneagon or 9-gon. Nonagon can be also defined as a two-dimensional closed figure having nine sides. The sides of a nonagon are always straight. The nonagon shape is two-dimensional. The figure of two types of nonagon is given below.

## Types of Nonagons

We can classify nonagons on the basis of two criteria. Nonagons are classified as Regular and Irregular on the basis of the length of sides while “Convex” and “Concave” are on the basis of interior angles.

• ### Regular Nonagon

The nine sides of a regular nonagon contain equal length. Each interior angle measures 140° and each exterior angle of 40°. It has a rotational equilibrium of order nine and nine lines of symmetry. The shape of a regular nonagon is given below:

• ### Irregular Nonagon

In the irregular type of nonagon, all sides, and all angles are not equal but the sum of all its interior angles always remains constant, that is 1260°. Some examples of irregularly shaped nonagon are given below.

• ### Convex Nonagon

The convex nonagon has its nine vertices pointing outwards. None of the interior angles measures more than 180°. All the diagonals of a convex nonagon lie inside the closed figure. A convex nonagon can be regular or irregular.

• ### Concave Nonagon

If a nonagon has at least one vertex pointing inwards with an interior angle greater than 180° is called a concave nonagon. In this type of nonagon at least one diagonal lies outside the closed figure. All concave polygons are always irregular in shape. The diagram of the convex nonagon is given below.

## Sides of Nonagons

A nonagon is made from nine straight sides and nine vertices that connect these sides. The name of nonagon denotes that it has nine sides since nona means nine and gon means sides. In the following figure AB, BC, CD, DE, EF, FG, GH, HI, and AI are the nine sides.

## Diagonals of a Nonagon

The number of diagonals of a nonagon is equal to 27. These diagonals are drawn in the following diagram.

## Interior Angles of a Regular Nonagon

A nonagon where all sides are equal in length is called a regular nonagon. The measure of each interior angle of a regular nonagon is equal to 140 degrees as shown in the figure.

The sum of all its interior angles is always constant i.e. 1260° even if the sides of the nonagon are not equal. Hence the measure of each internal angle of a regular nonagon is 140° and the measure of the central angle is 40° when it is divided into triangles as shown in the above diagram.

## Perimeter and Area of a Nonagon

A regular nonagon can be divided into 9 isosceles triangles with equal area, which means 9 congruent isosceles triangles.

The perimeter of a regular nonagon can be calculated as 9a where a is the side of a nonagon.

The area of a regular nonagon is calculated by the formula; A= $\frac{9}{4}$a2cot$\frac{π}{9}$; where a is the side of the nonagon.

Area can be found by A= $\frac{9}{2}$ar = 9r2tan$\frac{π}{9}$ ; where r is the incircle radius.

Or

A= $\frac{9}{2}$R2sin$\frac{2π}{9}$; where R is the circumcircle radius.

## Properties of a Regular Nonagon

• All the nine sides of a regular nonagon are equal.

• All the nine interior angles are equal and are 140o each and the sum of all interior angles is 1260o.

• All the nine exterior angles are equal and are 40o each.

• It has 27 diagonals.

• The number of triangles in the regular nonagon is 9.

• The perimeter of the nonagon is calculated as 9a, where a is the side length of the nonagon.

• All regular nonagons are convex nonagons.

## Did You Know?

A circle is not considered a polygon although it is also a plane figure due to its curvy shape and does not have sides or angles. Hence we can say that all the polygons are two dimensions in shapes but not all the two-dimensional figures are polygons.

## Problems on Nonagon

Some examples of problems based on nonagon are given below.

Example 1: Calculate the area of a regular nonagon of each side of 18 cm.

Solution: As we know, A = $\frac{9}{4}$a2cot$\frac{π}{9}$

A=2.25(18)22.747

A= 2.253242.747

A= 2,002.56 square cm

Hence the area of a regular nonagon will be 2,002.56 square cm.

Example 2: If the radius of the inscribed circle is 30 cm then find out the area and perimeter of a regular nonagon

Solution: It is given in the problem statement that radius (r) is 30 cm.

No of sides of a nonagon, n=9

Area, A = $\frac{9}{4}$a2cot$\frac{π}{9}$, where a is the side of the nonagon.

The length of the side of the nonagon is not given in the problem hence we will calculate its side first with the help of the following figure. Let’s consider one triangle from nonagon. Then the side of the polygon, a=2rtan$\frac{180}{9}$.

a=230 tan20o

a= 600.36397

a= 21.838

Now put the value of a in formula; A = $\frac{9}{4}$a2cot$\frac{π}{9}$

A= 2.25(21.838)22.747

A= 2947.588 square cm.

Perimeter of nonagon= 9a= 921.838=196.542 cm.

Hence the area of the nonagon is 2947.588 square cm and the perimeter is 196.542 cm.

Example 3: Find out the perimeter of a regular nonagon where the nonagon is inscribed in a circle of radius 26 cm.

Solution: The nonagon is inscribed in a circle and the radius is 26 cm. If the side of the polygon is a and radius of a circle is r then, a=2rtan$\frac{180}{9}$

a=226tan200

a=520.36397

a=18.926

Perimeter of nonagon=9a

=918.926

=170.334

Hence the perimeter of the nonagon is 170.334 cm.

## Practice Questions

1. Find out the perimeter of a regular nonagon where the nonagon is inscribed in a circle of radius 16 cm.

2. Calculate the area of a regular nonagon of each side of 2 cm.

## Conclusion

Nonagon shape is observed in daily life as an example of an irregular nonagon is the US Steel Building in Pittsburgh, Pennsylvania. We always encounter such problems where we are asked to calculate the area and perimeter of the nonagon. So this article would be quite helpful for understanding each aspect of a nonagon.

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## FAQs on Nonagon: Definition, Interior Angles and Properties

1. What is a 9-sided figure called?

A nine-sided figure is called a nonagon in geometry. It is a polygon that contains nine angles along with nine sides. The word nonagon comes from the word Latin “nonus” which means nine and the Greek word gon which means sides. Another name of nonagon is ‘enneagon’ which is derived from the Greek word ‘enneagonon’.

Nonagon can be also defined as a two-dimensional closed figure having nine sides. The nonagon sides are always straight. A nonagon shape is a two-dimensional shape.

2. What are the properties of nonagon?

The nonagon has some properties. These are related to interior, exterior angles, and sides.

• A nonagon has nine straight sides and nine vertices that are the connection of these sides.

• The sum of all the interior angles of a nonagon is equal to 1260 degrees.

• All the nine sides of a regular nonagon are equal and all the nine interior angles are equal and are 140o each.

•  All the nine exterior angles are equal and are 40o each.

3. What is a 10 sided shape called?

A 10 sided polygon is known as a decagon. We know that the interior angles are equal and are 144o each and the sum of all interior angles is 1440o, all the ten exterior angles are equal and are 36o each.