# Nonagon
A nonagon is a polygon with nine sides and nine angles, also called an enneagon or 9-gon.[1](https://www.mathopenref.com/nonagon.html) The name combines the Latin *nonus* (ninth) with the Greek-derived *-gon*; it dates from the mid 17th century, and the fully Greek alternative "enneagon" is sometimes preferred for consistency, though "nonagon" is simpler to spell and pronounce.[2](https://www.oxfordlearnersdictionaries.com/us/definition/english/nonagon) • [3](https://mathworld.wolfram.com/Nonagon.html) Nonagons may be regular or irregular, convex or concave.
| Key fact | Detail |
|---|---|
| Sides and angles | 9 sides, 9 angles; sum of interior angles is 1260° for every nonagon[1](https://www.mathopenref.com/nonagon.html) |
| Regular nonagon angles | Each interior angle 140°, each exterior angle 40°[1](https://www.mathopenref.com/nonagon.html) |
| Diagonals | 27, from the formula ½n(n − 3) with n = 9[1](https://www.mathopenref.com/nonagon.html) |
| Triangulation | Diagonals from one vertex divide it into 7 triangles (n − 2)[1](https://www.mathopenref.com/nonagon.html) |
| Area (regular) | \( A = \tfrac{9}{4} a^{2} \cot(\pi/9) \approx 6.182\,a^{2} \), where a is the side length[4](https://www.geeksforgeeks.org/maths/what-is-nonagon/) • [1](https://www.mathopenref.com/nonagon.html) |
| Perimeter (regular) | 9a, nine times the side length[4](https://www.geeksforgeeks.org/maths/what-is-nonagon/) |
| Constructibility | The regular nonagon cannot be constructed with compass and straightedge alone[5](https://math.stackexchange.com/questions/5099584/constructibility-of-a-regular-nonagon-and-the-possibility-of-constructing-an-80) |
## Angles and counting properties
The sum of the interior angles of any nonagon, regular or not, is 180° × (9 − 2) = 1260°. In a regular nonagon, where all sides and angles are equal, each interior angle is 1260° ÷ 9 = 140°, and each exterior angle, which forms a linear pair with the interior angle, is 180° − 140° = 40°.[1](https://www.mathopenref.com/nonagon.html) • [4](https://www.geeksforgeeks.org/maths/what-is-nonagon/) The exterior angles of any convex polygon sum to 360°, so nine exterior angles of 40° account for the full turn.[6](https://dfrrh0itwp1ti.cloudfront.net/nonagon)
Counting across the vertices gives two further fixed numbers. A diagonal joins two non-adjacent vertices; each of the 9 vertices sends 6 diagonals, and counting each diagonal once gives ½ × 9 × 6 = 27 diagonals in total.[6](https://dfrrh0itwp1ti.cloudfront.net/nonagon) Drawing all diagonals from a single vertex divides the nonagon into 7 triangles, and since each triangle contributes 180°, this reproduces the 1260° angle sum.[1](https://www.mathopenref.com/nonagon.html)
## Types of nonagon
A regular nonagon has nine equal sides and nine equal angles, making it symmetric and convex.[4](https://www.geeksforgeeks.org/maths/what-is-nonagon/) An irregular nonagon has sides and angles of different measures but keeps the same 1260° interior angle sum.[6](https://dfrrh0itwp1ti.cloudfront.net/nonagon) Irregular nonagons split into two classes: a convex nonagon has all interior angles below 180°, while a concave nonagon has one or more interior angles above 180°, producing an indented corner. A regular nonagon is always convex.[6](https://dfrrh0itwp1ti.cloudfront.net/nonagon)
## Area and perimeter
For a regular nonagon with side length a, the perimeter is simply 9a.[4](https://www.geeksforgeeks.org/maths/what-is-nonagon/) The area follows from the general regular-polygon formula in terms of the side length and the number of sides:
\[ A = \frac{9}{4}\, a^{2} \cot\!\left(\frac{\pi}{9}\right) \approx 6.182\, a^{2}. \]
The cotangent factor comes from assembling nine isosceles triangles with apex angle 40° around the center.[4](https://www.geeksforgeeks.org/maths/what-is-nonagon/) • [1](https://www.mathopenref.com/nonagon.html)
## Why the regular nonagon cannot be constructed
The regular nonagon is the second regular polygon, after the heptagon, that the ancient Greeks could not construct with ruler and compass; the triangle, square, hexagon, and octagon were within their reach.[7](https://www.edimast.it/index.php/edimast/article/download/70/60/202) The obstruction is visible in the central angle: each side of a regular nonagon subtends 360° ÷ 9 = 40° at the center, so constructing the polygon is equivalent to constructing a 40° angle, which means trisecting the 120° angle of a regular hexagon.[7](https://www.edimast.it/index.php/edimast/article/download/70/60/202) • [8](https://www.ms.uky.edu/~sohum/ma330/files/eqns_3.pdf)
Two classical results make this impossibility precise. Pierre Wantzel proved in 1837 that a cubic polynomial with rational coefficients and no rational root has no constructible roots; since cos 40° satisfies the cubic \( 4x^{3} - 3x + \tfrac{1}{2} = 0 \), which has no rational solutions, the 40° angle is not constructible and neither is the nonagon.[8](https://www.ms.uky.edu/~sohum/ma330/files/eqns_3.pdf) The same conclusion follows from the Gauss–Wantzel theorem: a regular n-gon is constructible if and only if n is a power of 2 times a product of distinct Fermat primes, primes of the form \( 2^{a} + 1 \) such as 3, 5, 17, 257, and 65537. For the nonagon, n = 9 = 3², and the criterion requires distinct Fermat primes, not a repeated one, so 9 fails the test.[5](https://math.stackexchange.com/questions/5099584/constructibility-of-a-regular-nonagon-and-the-possibility-of-constructing-an-80) • [9](https://people.math.sc.edu/mcnulty/algebrafirst.pdf) Some historians of mathematics claim that the angle trisection problem itself was first posed by the attempt to construct the regular nonagon.[7](https://www.edimast.it/index.php/edimast/article/download/70/60/202)
## Approximate and alternative constructions
Although exact compass-and-straightedge construction is impossible, close approximations exist. Dixon (1991) gives constructions for angles near the nonagonal 40°, including \( \tan^{-1}(5/6) \approx 39.805571^{\circ} \) and \( 2\tan^{-1}\!\bigl((\sqrt{3}-1)/2\bigr) \approx 40.207819^{\circ} \).[10](https://mathworld.wolfram.com/RegularNonagon.html) Conway and Guy (1996) give an exact neusis construction, a classical technique that allows marked straightedge placements and is based on angle trisection.[10](https://mathworld.wolfram.com/RegularNonagon.html)
Paper folding also succeeds where compass and straightedge fail. Origami can solve cubic equations, so it can trisect angles and fold a regular nonagon exactly from a square of paper; the key fold places two points onto two lines at once, producing a common tangent of two parabolas whose slope is \( \cos(2\pi/9) \).[11](https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv23n4.pdf) Madachy (1979) illustrates a simpler method of folding and knotting a strip of paper into a nonagon.[10](https://mathworld.wolfram.com/RegularNonagon.html)
## Nonagons in architecture
The nine-sided plan appears most prominently in the Bahá'í House of Worship in [Wilmette, Illinois](https://www.edgepedia.org/wilmette-illinois), completed in 1953 and designed by Louis Bourgeois. All nine Bahá'í houses of worship in the world are nine-sided, because Bahá'ís regard the number nine as a symbol of oneness and unity; the Wilmette temple has nine entrances, nine exterior pillars, and nine verses above its doors.[12](https://www.architecture.org/online-resources/buildings-of-chicago/baha-i-house-of-worship) In ancient Greek architecture no regular-nonagon buildings are known; the closest related structure is the Philippeion at Olympia, whose 18 Ionic columns form a regular octadecagon constructed by dividing the sides of a regular nonagon.[7](https://www.edimast.it/index.php/edimast/article/download/70/60/202) The nonagonal plan has also been used in modern work, from a nine-sided timber-frame hempcrete building in [Ashland, Oregon](https://www.edgepedia.org/ashland-oregon), to conceptual skyscraper designs built around a nonagonal rotating floor arrangement.[13](https://www.nextgenesisdesign.com/projects/kirby-nonagon) • [14](https://repositum.tuwien.at/bitstream/20.500.12708/187322/1/Proempers%20Martin%20-%202023%20-%20Nonagonal%20rotating%20tripple%20helix%20Innovative%20skyscraper...pdf)
## References
1. [Nonagon, Math Open Reference](https://www.mathopenref.com/nonagon.html)
2. [nonagon noun, Oxford Advanced Learner's Dictionary](https://www.oxfordlearnersdictionaries.com/us/definition/english/nonagon)
3. [Nonagon, Wolfram MathWorld](https://mathworld.wolfram.com/Nonagon.html)
4. [Nonagon: Properties, Types and Examples, GeeksforGeeks](https://www.geeksforgeeks.org/maths/what-is-nonagon/)
5. [Constructibility of a regular nonagon, Mathematics Stack Exchange](https://math.stackexchange.com/questions/5099584/constructibility-of-a-regular-nonagon-and-the-possibility-of-constructing-an-80)
6. [Nonagon, Cliffs-notes-style geometry reference](https://dfrrh0itwp1ti.cloudfront.net/nonagon)
7. [The regular nonagon, EDiMaST Volume 6 (2020), S. Gountouvas](https://www.edimast.it/index.php/edimast/article/download/70/60/202)
8. [Trigonometry equations and constructibility, University of Kentucky course notes (MA330)](https://www.ms.uky.edu/~sohum/ma330/files/eqns_3.pdf)
9. [Constructions by Straightedge and Compass, Lecture 21, University of South Carolina algebra notes](https://people.math.sc.edu/mcnulty/algebrafirst.pdf)
10. [Regular Nonagon, Wolfram MathWorld](https://mathworld.wolfram.com/RegularNonagon.html)
11. [Folding the Regular Nonagon, R. Geretschläger, Crux vol. 23 no. 4, Canadian Mathematical Society](https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv23n4.pdf)
12. [Bahá'í House of Worship, Chicago Architecture Center](https://www.architecture.org/online-resources/buildings-of-chicago/baha-i-house-of-worship)
13. [Kirkby Nonagon project, NextGenesisDesign](https://www.nextgenesisdesign.com/projects/kirby-nonagon)
14. [Nonagonal rotating triple helix: innovative skyscraper design, TU Wien repository (2023)](https://repositum.tuwien.at/bitstream/20.500.12708/187322/1/Proempers%20Martin%20-%202023%20-%20Nonagonal%20rotating%20tripple%20helix%20Innovative%20skyscraper...pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Elementary and Euclidean geometry › Plane figures*
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