What is a Nonagon?
A closed two dimensional shape having nine sides is called a nonagon. So, a nonagon is a polygon with nine sides, nine vertices and nine interior angles. Like any other polygon, the sum of exterior angles of a nonagon is equal to 360°.
The image represents a nonagon.
We can spot objects around us that are ‘nonagonal’ in shape, such as some coins, glass tumblers or dishes.
Types of Nonagons
Depending on the measurement of the sides, a nonagon can be of two types:
a. Regular Nonagon
b. Irregular Nonagon
A regular nonagon has all nine sides that are equal in length. But in an irregular nonagon the nine sides are not equal in length. The two types of nonagon are presented in the image.
Angle Measures in a Nonagon
Let’s look at the angle measures of a nonagon by applying the standard polygon formulas.
- \(Sum of interior angles = 180^{\circ }\left( n – 2 \right)\)
\(= 180^{\circ } \left( 9 – 2 \right)\) [Substitute n = 9]
\(= 1260^{\circ } \) [Simplify]
So, the sum of the interior angles of a nonagon is 1260°.
- \(Each\text{ }interior\text{ }angle\text{ }of\text{ }a\text{ }regular\text{ }nonagon = \frac{Sum\text{ }of\text{ }interior\text{ }angles}{Number\text{ }of\text{ }sides}= \frac{1260^{\circ }}{9} = 140^{\circ }\)
So, each interior angle of a regular nonagon measures 140°.
- \(Each\text{ }exterior\text{ }angle\text{ }of\text{ }a\text{ }regular\text{ }nonagon = \frac{Sum\text{ }of \text{ }exterior \text{ }angles}{Number\text{ }of\text{ }sides} = \frac{360^{\circ }}{9} = 40^{\circ }\)
So, each exterior angle of a regular nonagon measures 40°.
[Note: For a given vertex, the interior and the exterior angle of a nonagon form a linear pair of angles.]
Solved Examples
Example 1: Draw the diagonals of a nonagon. How many diagonals does a nonagon have?
Solution:
Diagonals are formed by connecting two vertices of any shape that do not lie on the same side. So the diagonals of a nonagon can be drawn as in this image:
Now count the number of diagonals. There are 27 diagonals.
Hence, a nonagon has 27 diagonals.
Example 2: Identify the type of nonagon presented in the following.
Solutions:
All four shapes have nine sides with different side lengths.
Hence, all four shapes are irregular nonagons.
Example 3: Find the measure of x.
Solution:
Here, x is the measure of the interior angle and 135° is the exterior angle, both formed at the same vertex of the polygon. So, these two angles form a linear pair.
x + 135° = 180° [Linear pair]
x = 180° – 135° [Substitute value]
x = 45° [Subtract 135° on both sides]
So, the measure of x is 45°.
Any 2D shape that has nine sides is in the shape of a nonagon, and is usually referred to as being ‘nonagonal in shape’.
Like any other polygon, the exterior angles of an irregular nonagon add up to 360°.
A shape with seven sides is called a heptagon.
A circle does not have any sides that are straight lines, so it is not a polygon.