A polygon is a plane shape with straight sides.
Is it a Polygon?
Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).
|Not a Polygon|
(has a curve)
|Not a Polygon|
(open, not closed)
Polygon comes from Greek. Poly- means "many" and -gon means "angle".
Types of Polygons
Regular or Irregular
A regular polygon has all angles equal and all sides equal, otherwise it is irregular
Concave or Convex
A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.
If any internal angle is greater than 180° then the polygon is concave. (Think: concave has a "cave" in it)
Simple or Complex
A simple polygon has only one boundary, and it doesn’t cross over itself. A complex polygon intersects itself! Many rules about polygons don’t work when it is complex.
(this one’s a Pentagon)
(also a Pentagon)
|Irregular Hexagon||Concave Octagon||Complex Polygon|
(a "star polygon",
in this case a pentagram )
Play With Them!
Try Interactive Polygons … make them regular, concave or complex.
Names of Polygons
|If it is a Regular Polygon…|
|Triangle (or Trigon)||3||60°|
|Quadrilateral (or Tetragon)||4||90°|
|Heptagon (or Septagon)||7||128.571°|
|Nonagon (or Enneagon)||9||140°|
|Hendecagon (or Undecagon)||11||147.273°|
|n-gon||n||(n-2) × 180° / n|
You can make names using this method:
Example: a 62-sided polygon is a Hexacontadigon
BUT, for polygons with 13 or more sides, it is OK (and easier) to write "13-gon", "14-gon" … "100-gon", etc.
Quadrilateral (4 Sides)
A Quad Bike has 4 wheels
Pentagon (5 Sides)
The "Pentagon" in Washington DC has 5 sides
Hexagon (6 Sides)
Honeycomb has Hexagons
Septagon (7 Sides)
Think Septagon is a "Seven-agon"
Octagon (8 Sides)
An Octopus has 8 tentacles
Nonagon (9 Sides)
Think Nonagon is a "Nine-agon"
Decagon (10 Sides)
Think Decagon has 10 sides,
just like our Decimal system has 10 digits
Elementary level (6th Grade)
What do you call a 13 sided polygon?
Thank you for your help.
There seems to be an “official”set of names for polygons, uniformly derivedfrom the Greek (after 4 sides).
After about 6 sides mathematicians usually refer to these polygons as n-gons. So your 13 sided polygon I would call a 13-gon.
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|Regular nonagon (enneagon)|
A regular nonagon (enneagon)
|Edges and vertices||9|
|Symmetry group||Dihedral (D9), order 2×9|
|Internal angle ( degrees )||140°|
|Properties||Convex , cyclic , equilateral , isogonal , isotoxal|
In geometry , a nonagon ( // ) or enneagon ( // ) is a nine-sided polygon or 9-gon.
The name “nonagon” is a prefix hybrid formation, from Latin (nonus, “ninth” + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century. The name “enneagon” comes from Greek enneagonon (εννεα, “nine” + γωνον (from γωνία = “corner”)), and is arguably more correct,  though less common than “nonagon”.
- 1 Regular nonagon
- 2 Construction
- 3 Approximations
- 3.1 Approximation I
- 3.2 Approximation II
- 4 Symmetry
- 5 Tilings
- 6 Graphs
- 7 Pop culture references
- 8 Architecture
- 9 See also
- 10 References
- 11 External links
Regular nonagon[ edit ]
A regular nonagon is represented by Schläfli symbol 9 and has internal angles of 140°. The area of a regular nonagon of side length a is given by
where the radius r of the inscribed circle of the regular nonagon is
and where R is the radius of its circumscribed circle :
Construction[ edit ]
Although a regular nonagon is not constructible with compass and straightedge (as 9 = 32, which is not a product of distinct Fermat primes ), there are very old methods of construction that produce very close approximations. 
It can be also constructed using neusis , or by allowing the use of an angle trisector .
Nonagon, an animation from a neusis construction based on the angle trisection 120° by means of the Tomahawk
Approximations[ edit ]
Approximation I[ edit ]
- Accuracy (linear): 10-6
The following is an approximate construction of a nonagon using a straightedge and compass .
Example to illustrate the error, when the constructed central angle is 39.99906°:
At a circumscribed circle radius r = 100 m, the absolute error of the 1st side would be approximately 1.6 mm.
Approximation II[ edit ]
- Accuracy (linear): 10-10
- Downsize the angle JMK (also 60°) with four bisections of angle and make a thirds of circular arc MON with an approximate solution between bisections of angle w3 and w4.
- Straight auxiliary line g aims over the point O to the point N (virtually a ruler at the points O and N applied), between O and N, therefore no auxiliary line.
- Thus, the circular arc MON is freely accessible for the later intersection point R.
- RMK = 40.0000000052441…°
- 360° ÷ 9 = 40°
- RMK – 40° = 5.2…E-9°
- Example to illustrate the error:
- At a circumscribed circle radius
- r = 100,000 km, the absolute error of the 1st side would be approximately 8.6 mm.
See also the calculation (Berechnung, German) .
Nonagon at a given circumscribed circle , animation]]
Nonagon at a given side length, animation]]
Symmetry[ edit ]
Symmetries of a regular enneagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.
The regular enneagon has Dih9 symmetry , order 18. There are 2 subgroup dihedral symmetries: Dih3 and Dih1, and 3 cyclic group symmetries: Z9, Z3, and Z1.
These 6 symmetries can be seen in 6 distinct symmetries on the enneagon. John Conway labels these by a letter and group order.  Full symmetry of the regular form is r18 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g9 subgroup has no degrees of freedom but can seen as directed edges .
Tilings[ edit ]
The regular enneagon can tessellate the euclidean tiling with gaps. These gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called H(*;3;*;) with H representing *632 hexagonal symmetry in the plane.
Graphs[ edit ]
The K9 complete graph is often drawn as a regular enneagon with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex .
Pop culture references[ edit ]
They Might Be Giants have a song entitled “Nonagon” on their children’s album Here Come the 123s . It refers to both an attendee at a party at which “everybody in the party is a many-sided polygon” and a dance they perform at this party.  Slipknot ‘s logo is also a version of a nonagon, being a nine-pointed star made of three triangles. King Gizzard & the Lizard Wizard have an album titled ‘ Nonagon Infinity ‘, the album art featuring a nonagonal complete graph.
Architecture[ edit ]
Temples of the Baha’i Faith are required to be nonagonal.
The U.S. Steel Tower is an irregular nonagon.
See also[ edit ]
- Enneagram (nonagram)
- Trisection of the angle 60°, Proximity construction
References[ edit ]
- ^ Eric W. Weisstein . “Nonagon” . > MathWorld –A Wolfram Web Resource. Retrieved 24 October 2018.
- ^ J. L. Berggren, “Episodes in the Mathematics of Medieval Islam”, p. 82 – 85 Springer-Verlag New York, Inc. 1st edition 1986, retrieved on 11 December 2015.
- ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss , (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
- ^ TMBW.net
External links[ edit ]
- Properties of a Nonagon (with interactive animation)
- 9 (number)
- Articles with short description
- This page was last edited on 29 November 2018, at 12:43 (UTC).
- Text is available under the Creative Commons Attribution-ShareAlike License ;
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