Nonagon, a polygon with nine sides and nine angles, is a geometric shape that captures mathematical intrigue and visual diversity. Whether regular or irregular, convex or concave, the nonagon takes on various forms based on the equality of its sides and the measurements of its interior angles.

In this article, we will learn the meaning and definitions, types, properties, area and perimeter of a nonagon, and angles of a nonagon, along with some solved examples.

Nonagon Shape

A nonagon shape is a polygon with nine sides. It is formed by connecting nine straight line segments, also known as sides, between nine corners or vertices. The nonagon can take different forms, categorized as regular or irregular, convex or concave, based on the equality of its sides and the measurements of its interior angles.

Definition of Nonagon

A nonagon is a polygon with nine sides and nine angles, and its form can be either regular or irregular. The sides connect the vertices, and the shape can be convex or concave based on the equality of its sides and angles.

Read More about Regular Polygon

Sides of Nonagon

A nonagon has nine sides. The sides of a nonagon are the straight lines that connect the corners of the shape. These nine sides can be of the same length or different lengths. A nonagon is a geometric figure with 9 sides, and it can take different forms such as regular, irregular, convex, or concave, based on the lengths of its sides and the angles inside the shape.

Types of Nonagon

A nonagon is divided into 4 different types based on their sides, angles, and vertices.

• Regular Nonagon or Simple Nonagon
• Irregular Nonagon or Complex Nonagon
• Convex Nonagon
• Concave Nonagon

Regular Nonagon

A regular nonagon is a polygon with nine equal sides and nine equal angles. In simple terms, it’s a shape with nine straight sides that are all the same length and nine angles that are all the same size. This makes a regular nonagon a symmetrical and evenly proportioned geometric figure. Each interior angle of a regular nonagon measures 140 degrees, and the sum of all its interior angles is always 1260 degrees.

Properties of Regular Nonagon

A regular Nonagon possesses the following properties:

• A regular nonagon is a polygon with nine equal sides.
• It has nine equal interior angles.
• Each interior angle in a regular nonagon measures 140 degrees.
• The sum of all interior angles in a regular nonagon is always 1260 degrees.
• The exterior angles of a regular nonagon are also equal.
• It is a symmetrical and evenly proportioned geometric shape.
• In contrast to irregular nonagons, all sides and angles in a regular nonagon are identical.
• Regular nonagons exhibit a high degree of geometric regularity, making them distinct from nonagons with varying side lengths and angle measures.

Irregular Nonagon

An irregular nonagon is a polygon with nine sides and nine angles, but unlike a regular nonagon where all sides and angles are equal, in an irregular nonagon, the sides and angles have different lengths and measures.

Determining the properties of an irregular nonagon involves breaking it down into smaller polygons, such as triangles and quadrilaterals, and analyzing these smaller parts individually. Irregular nonagons do not have a consistent pattern in terms of side lengths and angles, making their properties more diverse compared to regular nonagons.

Convex Nonagon

A convex nonagon is a polygon with nine sides, and it has the characteristic that all its interior angles are less than 180 degrees. In simpler terms, a convex nonagon is a shape with nine straight sides that don’t curve inward.

The angles inside this polygon point outward, and there are no “dents” or inward-pointing corners. Convex nonagons have a regular and smooth appearance without any angles bending inward.

Concave Nonagon

A concave nonagon is a polygon with nine sides and nine angles, and it has at least one interior angle that measures more than 180 degrees. In simpler terms, it has a “caved-in” or indented part, making the shape somewhat hollow or concave.

Picture it like a nonagon with one or more “dents” on the inside. This is different from a convex nonagon, where all interior angles are less than 180 degrees and the shape doesn’t have any indented parts.

Nonagon Angles

A nonagon consists of nine angles. It has both interior and exterior angles. The interior angles are the angles inside the nonagon, and there are nine of them. Similarly, there are nine corresponding exterior angles.

Interior Angles of a Nonagon

The interior angles of a nonagon, a polygon with nine sides, are the angles formed inside it. To find the measure of each interior angle, you can use the formula:

Interior angle = [(n-2) × 180] / n

For a nonagon, ( n = 9 ). Substituting this value into the formula:

Interior angle = [(9-2) × 180]/9 = 140

So, each interior angle of a regular nonagon measures 140°.

Sum of Interior Angles of a Nonagon

The sum of the interior angles of a nonagon, whether regular or irregular, is always 1260°. This can be calculated using the formula:

Sum of interior angles = (n-2) × 180°

Substituting (n = 9), we get

Sum of interior angles = (9-2) × 180 = 1260

This means that the total of the interior angles in a nonagon is 1260 degrees.

Read more, Sum of Interior Angle of Polygon

Exterior Angles of a Nonagon

The exterior angles of a nonagon, which is a polygon with nine sides, are the angles formed outside of it. To find the measure of each exterior angle, you can use the formula:

Exterior angle = 360/n

For a nonagon, n = 9 . Substituting this value into the formula, we get:

Exterior angle = 360/9 = 40

Therefore, each exterior angle of a nonagon measures 40°.

In the context of the image provided, it shows that the exterior angle and the interior angle together form a straight line, meaning they add up to 180°. Consequently, each exterior angle of a regular nonagon is 40°, and the sum of all exterior angles is 360°.

Diagonals of a Nonagon

A diagonal is a line segment that connects two vertices of a polygon but is not next to them. In a nonagon, which is a polygon with nine sides, there are 27 diagonals. To figure this out, you can use a formula:

Number of diagonals = 1/2 × n × (n −3)

Where n is the number of sides of the polygon.

For a nonagon (n=9), you substitute the value into the formula: Number of diagonals = 1/2 × 9 × (9−3)

= 1/2 × 9 × 6

= 27

∴ A nonagon has 27 diagonals.

How Many Sides are in a Nonagon?

We know that, Nonagon is polygon with 9 sides. Thus, there are 9 sides in a nonagon.

Perimeter of a Nonagon

The perimeter of a polygon is the total length around its edges. To determine the perimeter of a nonagon, which is a polygon with nine sides, it required to know the length of each of its 9 sides. The perimeter is found by adding up the lengths of all the sides. In the case of a regular nonagon, where all sides are of equal length, you can find the perimeter by multiplying the length of one side by 9. This can be expressed as:

Perimeter of a nonagon = Sum of all its sides

Perimeter of a regular nonagon = 9 times the length of one side (denoted as ‘a’ for the nonagon)

Example: What is the total length of the sides of a nonagon if each side measures 4 units?

Solution:

As we know the formula is:

Perimeter of Nonagon = 9 × Side length

Perimeter = 9 × 4 = 36 units

Area of Nonagon

The area of a nonagon, a polygon with nine sides and nine angles, represents the space it encloses. This measurement is expressed in square units, like square meters or square feet.

For a regular nonagon, where all sides and angles are equal, the formula for its area is given by:

[Tex]\bold{\text{Area} = \frac{9}{4} \times a^2 \times \cot\left(\frac{\pi}{9}\right)}[/Tex]

Here, ‘a’ is the length of one side of the nonagon, and π represents the mathematical constant, approximately equal to 3.14.

In the case of an irregular nonagon, which has sides and angles of varying lengths and measures, determining its area involves breaking it down into smaller polygons, like triangles and quadrilaterals. The areas of these smaller parts can then be calculated using known formulas, and the results are added together to find the total area of the irregular nonagon.

Example: Find the area of a nonagon with side length 5 units.

Area of a nonagon with a side length of 5 units is calculated using the formula:

Area = 9/4 × (side length)² × cot(π/9)

On substituting the given side length (5 units) in the formula:

⇒ Area = 9/4 × 5² × cot(π/9)

⇒ Area = 9/4 × 25 × cot(π/9)

⇒ Area = 225/4 × cot(π/9) sq units

How to Draw a Nonagon?

Follow the given below steps to create a nonagon

Step 1: Begin by placing a dot on your paper. This will be the center of your nonagon.

Step 2: Use a compass to draw a circle around the dot. This circle will be the circumcircle of the nonagon.

Step 3: With a protractor, mark nine equally spaced points around the circumference of the circle. Each point represents a vertex of the nonagon.

Step 4: Use a ruler to connect the center dot to each of the nine points on the circumference. These lines form the sides of the nonagon.

Step 5: Connect the last point to the starting point to close the figure.

Step 6: If you used any construction lines or extra marks, erase them to leave a clean nonagon shape.

Formulas of Nonagon

The basic formulas of the Nonagon are,