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 ninesided 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 nonadjacent 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 × (n3), 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 × (n3) = 1/2 × 9 × (93) = 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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Examples on Nonagon

Example 1: Find the sum of the interior angles of a nonagon using the appropriate formula.
Solution: The sum of the interior angles of a nonagon 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°. Therefore, the sum of interior angles of a nonagon is 1260°.

Example 2: Find the perimeter of a regular nonagon whose one side is 6 units.
Solution: We can find the perimeter of the nonagon using the formula, Perimeter of a regular nonagon = 9a, where a = 6 units. After substituting this value in the formula, we get, Perimeter of a regular nonagon = 9a = 9 × 6 = 54 units. Therefore, the perimeter of the nonagon is 54 units.

Example 3: State true or false:
a.) A concave nonagon is one in which at least one interior angle is a reflex angle.
b.) Each interior angle of a regular nonagon measures 110°.
Solution:
a.) True, a concave nonagon is one in which at least one interior angle is a reflex angle.
b.) False, each interior angle of a regular nonagon measures 140°.
FAQs on Nonagon
What is a Nonagon Shape?
Nonagon is a polygon with 9 sides, 9 vertices and 9 interior angles. A nonagon can be regular, irregular, convex or concave depending upon its sides and interior angles.
What is a Regular Nonagon?
A regular nonagon is one in which all the sides are of equal length and all the interior angles are of equal measure. This means each interior angle of a regular nonagon is equal to 140°, and therefore, each respective exterior angle is 40°.
What is the Sum of the Interior Angles of a Nonagon?
The sum of the interior angles of a nonagon is 1260°. This can be calculated with the help of the formula, Sum of interior angles of a polygon = (n  2) × 180°, where 'n' = number of sides of the polygon. After substituting the value of n = 9, we get, Sum of interior angles of a polygon = (9  2) × 180° = 1260°.
What is the Interior Angle of a Nonagon?
The interior angle of a nonagon is formed inside the nonagon. Each interior angle of a regular nonagon measures 140°. This can be calculated using the formula, Interior angle of a regular polygon =\(\frac{180n–360} {n}\), where n = the number of sides of the polygon. Since a nonagon has 9 sides, n = 9. After substituting this value of 'n' in the formula, we get \(\begin{align} \frac{180(9)–360} {9} = 140^\circ \end{align}\)
What is the Exterior Angle of a Nonagon?
Each exterior angle of a regular nonagon measures 40°. When any side of a nonagon is extended outwards, the angle that is formed with this side and its adjacent side is called the exterior angle of the nonagon. Since each exterior angle forms a linear pair with its corresponding interior angle, it means they both sum up to 180°. Therefore, each exterior angle of a regular nonagon is, 180°  140° = 40°. The sum of the exterior angles of a nonagon is 360°.
How many Sides are there in a Nonagon?
There are 9 sides in a nonagon. A nonagon can be regular, irregular, convex, or concave depending on the length of its sides and angles.
How many Triangles are there in a Nonagon?
7 triangles can be drawn in a nonagon. A nonagon can be split into triangles with the help of its diagonals that are drawn from its vertices. The number of triangles that are created by these diagonals, can be calculated with the formula: (n  2), where n = the number of sides. In this case, n = 9. So, after substituting this value in the formula we get, 9  2 = 7. Therefore, 7 triangles can be formed in nonagon.
What is a Convex Nonagon?
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.
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