Nonagon: Types, Properties & Formulas

A nonagon is a polygon that has ninе sidеs and ninе anglеs. It bеlongs to thе family of rеgular polygons charactеrizеd by all sidеs having еqual lеngth and all interior anglеs bеing congruent.

• The word nonagon has been derived from the Latin word ‘nonus’, meaning nine, and the Greek word ‘gon’, meaning sides.
• It consists of 27 diagonals, and the sum of the interior angles is 1260°
• The two-dimensional figure forms various shapes based on the equality of its sides and the measurements of its interior angles.
• It is categorized into regular or irregular, convex or concave forms depending on whether their sides and angles are еqual or not. 
• You can check out various shapes of nonagon in real life, including glass tumblers or dishes.
• The U.S. Steel Tower in Pittsburgh is a structural example of an irregular nonagon that you can explore to understand the figure better.

Key Terms: Nonagon, Nonagon Shapes, Nonagon Angles, Nonagon Diagonals, Regular Nonagon, Properties of Nonagon, Irregular Nonagon, Types of Nonagon, Formulas of Nonagon

---

What is Nonagon?

[Click Here for Sample Questions]

Nonagon is a two-dimensional figure with nine sides, nine interior and nine outside angles.

• The regularity or irregularity of a nonagon form depends on its angles and sides. 
• In a convex nonagon, all internal angles are less than 180°.
• It is popular with the name Enneagon or 9-gon.
• Nonagon belongs to the class of concyclic points and equilateral polygons.
• If you draw the graph for the figure, it will give a projection of 9 vertices and 36 edges of the 8-simplex.

Also Read:

---

Nonagon Shape

[Click Here for Previous Year Questions]

A nonagon is a nine-sided geometric shape that can be regular, irregular, convex, or concave, depending on its sides and internal angles. It has nine vertices with an interior angle of 1260 degrees.

Nonagon

---

Types of Nonagon

[Click Here for Sample Questions]

There are four different types of nonagons, which are differentiated based on their sides, angles, and vertices.

Regular Nonagons

A regular nonagon is a type of nonagon which has all nine sides and all nine internal angles in the same measure.

• Conversely, a nonagon is called irregular when its sides have variable lengths and angles have different measurements.
• Regular nonagons possеss rotational symmetry, meaning thеy can bе rotatеd by a certain anglе and still appеar thе samе.
• It is also known as a simple nonagon. 
• A simple nonagon has no self-intersection which means its sides don't meet at any point.
• If you want to see in real life examples of regular polygons then you can check out Baháʼí Houses of Worship in Australia.

Properties of Regular Nonagon

The properties of Regular Nonagon are as follows:

• A regular nonagon is a polygon with nine equal sides and nine equal interior angles.
• The sum of each interior angle is equal to 140 degrees.
• The sum of each interior angle is equal to 360 degrees.
• Regular nonagons showcase a high degree of geometric regularity, making them unique compared to nonagons with varying side lengths and angle measures.

Regular Nonagon

Irregular Nonagon

Irregular nonagon is a type of nonagon which has non-equal interior angles or sides. The total of the nine inner angles is 1260°. 

• Convex and concave irregular nonagons are also possible.
• It is also known as complex nonagons where sides are intersected.
• Nonagons with extra internal areas and self-intersections are called complex nonagons. 

Irregular Nonagon

Convex Nonagons

Any nonagon with all of its internal angles measuring less than 180° is said to be convex. Convex nonagons appear to jut outward because each inner angle is smaller than 180°. It can be both regular and irregular.

Convex Nonagon

Concave Nonagons

A nonagon that has at least one reflex angle inside of it is said to be concave. The unusual shape of a concave nonagon is caused by the possibility of its vertices being inward.

Concave Nonagon

---

Nonagon Angles

[Click Here for Previous Year Questions]

As you have studied in the above sections, a nonagon comprises nine interior and nine exterior angles, respectively. There are two nonagon angles, which are as follows:

Sum of interior angles of Nonagon

Each regular polygon's internal angle is given by the formula (180n-360)/n, where n is the polygon's number of sides. The interior angle of a regular polygon when n = 9 is given by 

• (180*9 – 360)/9
• Internal angle = 140°
• The sum of a polygon's interior angles, = (n-2)*180°, where n is the number of sides.
• Given a nonagon's number of sides, its total number of angles = (9-2)*180°
• 7*180°
• 1260°

One can also multiply the number of triangles in a polygon by 180° to determine the sum of the internal angles of the polygon. 

• Sum of a nonagon's internal angles = 7 *180° = 1260

Sum of exterior angles of Nonagon

A typical nonagon's exterior angles are all 40°. The exterior angles and inner angles combine to form a linear pair of angles, meaning their sum equals 180°. 

• A normal nonagon will therefore have exterior angles of 180° - 140° = 40°. 
• A nonagon's exterior angles add up to 360°.

---

Diagonals of Nonagon

[Click Here for Sample Questions]

In a nonagon, there are 27 diagonals. The pentagon's total number of diagonals can be computed by combining its non-adjacent vertices. 

• The formula for diagonals of a nonagon is as follows: 1/2 × n × (n-3), where n = number of sides in the polygon.
• In this case, n = 9. 
• Upon entering n = 9 into the calculation
• 1/2 × n × (n-3) = 1/2 × 9 × (9-3) = 27. 

---

Perimeter of Nonagon

[Click Here for Previous Year Questions]

The entire length of a polygon's boundary is its perimeter. The perimeter of a nonagon is equal to the sum of its sides.

• Since each side of a regular nonagon is the same length, one may find the perimeter by multiplying the length of one side by 9. 
• Mathematically, we can write it as perimeter = 9a , where 'a' denotes the length of a nonagon side.

---

Area of Nonagon

[Click Here for Sample Questions]

The area of a regular nonagon of side length a is given by:

Area = (9/4)a² cot(π/9)

Or

Area = (9/2)ar = 9r² tan(π/9)

• Here, r is the radius of the circle inscribed inside the regular nonagon and r = (a/2) cot(π/9).
• Hence, Area = (9/2)R² sin(2π/9)
• Here, R is the radius of the circumscribed circle of the regular nonagon.

---

Formulas of Nonagon

[Click Here for Previous Year Questions]

Important formulas of nonagon are as follows:

---

Things to Remember

• A nonagon is a two-dimensional figure (polygon) with nine sides and nine angles
• Regular, irregular, convex, and concave are its four main types. 
• The majority of the characteristics and equations only apply to normal nonagons, which have equal sides and equal exterior and interior angles of 140°and 40°, respectively. 
• All nonagons, however, have an interior angle sum of 1260 degrees overall.
• The nine sides of the polygon can either be the same length or different lengths. 

Also Read:

---

Previous Year Questions

1. Extraction of metal from the ore cassiterite involves...[JEE Advanced 2011]
2. Commonly used vectors for human genome sequencing are...[NEET UG 2014]
3. Interfascicular cambium and cork cambium are formed due to​..
4. Pneumotaxic centre is present in​...[UP CPMT 2007]
5. Reaction of HBr with propene in the presence of peroxide gives….[NEET UG 2004]
6. Assuming the expression for the pressure exerted by the gas on the walls of the container, it can be shown that pressure is...[MHT CET 2016]
7. Which among the following is the strongest acid?...[TS EAMCET 2017]
8. Isopropyl alcohol on oxidation forms​..
9. A vector is not changed if​..
10. Which of the following arrangements does not represent the correct order of the property stated against it?...[JEE Main 2013]
11. The major product of the following reaction is​...[JEE Main 2019]
12. Major product of the following reaction is..[JEE Main 2023]
13. The percentage of nitrogen in urea is about..
14. The electric field at a point is​
15. Which of the following statements is true?​..[JKCET 2006]