Definition, Formula, Examples | Nonagon Shape - Cuemath

Nonagon

Nonagon is a polygon that has 9 sides, 9 interior angles and 9 exterior angles. A nonagon has 27 diagonals and the sum of the interior angles of a nonagon is 1260°. A nonagon shape can be regular or irregular depending upon the sides and angles. Let us learn more about the nonagon shape.

1.	What is a Nonagon Shape?
2.	Nonagon Sides
3.	Regular Nonagon
4.	Sum of Interior Angles of a Nonagon
5.	FAQs on Nonagon

What is a Nonagon Shape?

A nonagon is a nine-sided geometrical figure which can be regular, irregular, convex or concave depending upon its sides and interior angles. Observe the figure given below to see what a nonagon looks like.

Nonagon Sides

As seen in the figure given above, a nonagon has 9 sides and the name 'Nonagon' was derived from the Latin words 'nonus' and 'gon' which mean a polygon with nine sides. These 9 sides of a nonagon can be equal or of different measures.

Regular Nonagon

A regular nonagon is one in which all the 9 sides are of equal length and the 9 interior angles are of equal measure. On the other hand, when the sides of a nonagon are of unequal lengths and the angles are of different measure, it is called an irregular nonagon.

Properties of a Regular Nonagon

The following points show the properties of a regular nonagon.

• In a regular nonagon, all the sides are of equal length, and all the interior angles are of equal measure.
• The sum of the interior angles of a regular nonagon is 1260° and the sum of the exterior angles is 360°.
• Each interior angle of a regular nonagon measures 140°.

Observe the following figure which shows a regular nonagon and an irregular nonagon.

Convex Nonagon and Concave Nonagon

A convex nonagon has the following properties.

• A convex nonagon is one in which all the interior angles measure less than 180°.
• Since each interior angle of a convex nonagon is less than 180°, convex nonagons seem to bulge outwards.

A concave nonagon has the following properties.

• A concave nonagon is one in which at least one interior angle is a reflex angle.
• In a concave nonagon, the vertices of a concave nonagon can be inwards which gives it that peculiar shape.

Observe the figure given below which shows a convex nonagon and a concave nonagon shape.

Nonagon Angles

There are 9 interior angles and 9 respective exterior angles in a nonagon. Observe the figure given below which shows the interior angles and the exterior angle of a regular nonagon. Let us read about them in detail.

Nonagon Interior Angles

Each interior angle of a regular nonagon measures 140°. This can be calculated using the formula for the interior angles of a regular polygon. Interior angle of a regular polygon =\(\frac{180n–360} {n}\), where n = the number of sides of the polygon. In a nonagon, n = 9. After substituting this value of 'n' in the formula, we get \(\begin{align} \frac{180(9)–360} {9} = 140^\circ \end{align}\)

Sum of Interior Angles of a Nonagon

The sum of the interior angles of a nonagon is always 1260°. This sum remains the same irrespective of the nonagon being a regular or an irregular nonagon and this can be calculated using the formula, Sum of interior angles of a polygon = (n - 2) × 180°, where 'n' = number of sides of the polygon. Here, after substituting the value of n = 9, we get, Sum of interior angles of a polygon = (9 - 2) × 180° = 1260°.

Nonagon Exterior Angles

Each exterior angle of a regular nonagon measures 40°. If we observe the figure given above, we can see that the exterior angle and interior angle form a linear pair of angles, which means they sum up to 180°. Hence, each exterior angle of a regular nonagon will be 180° - 140° = 40°. The sum of the exterior angles of a nonagon is 360°.

Properties of Nonagon

The following properties of a nonagon help to distinguish it from other polygons.

• A nonagon is a polygon with 9 sides, 9 interior angles, and 9 exterior angles.
• The interior angles of a nonagon sum up to 1260°
• A nonagon has 27 diagonals.

Nonagon Diagonals

There are 27 diagonals in a nonagon. These diagonals are drawn by joining its non-adjacent vertices and the total number of diagonals in a nonagon can be calculated using the formula, Number of diagonals in a polygon = 1/2 × n × (n-3), where n = number of sides in the polygon. Here, n = 9. After substituting this value of n = 9 in the formula we get, Number of diagonals in a polygon: 1/2 × n × (n-3) = 1/2 × 9 × (9-3) = 27. Therefore, a nonagon has 27 diagonals.

Perimeter of Nonagon

The perimeter of a polygon is the total length of its boundary. So, in order to find the perimeter of a nonagon, we need to know the length of all 9 sides. The sum of all the sides gives the perimeter of a nonagon. In case of a regular nonagon, all the sides are of equal length, so we can multiply the length of one side by 9 to get the perimeter. These formulas can be expressed as follows.

Perimeter of a nonagon = Sum of all its sides

Perimeter of a regular nonagon = 9a (Where 'a' is the length of one side of the nonagon)

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