What is a nonagon in math

Regular neunagon

What is a regular nine-corner?

The regular nine-corner is a polygon with
nine equally long sides and
nine interior angles of equal size.
The Neuneck also known as Nonagon or less often Enneagon.
In English are the names Nonagon or Enneagon common.

On this page, the regular nine-corner is usually simply called the nine-corner.

Sizes of the hexagon Top
Winkel im Neunagon


Formulas

Side and diagonals

Radius of the perimeter, radius of the inscribed circle, height

Is the side a given, the radius can be derived from it r of the inscribed circle, the radius R. of the perimeter, the diagonals d1, d2 and d3 and the height H, the area A. and the scope U calculate. As with the regular heptagon, there are hardly any root terms here. With the exception of sin60 °, you have to leave the number terms of trigonometric functions as they are.


Derivation of the formulas
Radius of the circumference and inscribed circle
The following applies: sin20 ° = (a / 2) / R or R = a / (2sin20 °)

The following applies: cot20 ° = r / (a ​​/ 2) or r= (a / 2) cot20 ° = (a / 2) (cos20 ° / sin20 °)


height
......
According to the set of centers, the circumferential angle at the top is half as large as the center angle (red marking).

In the yellow triangle cot10 ° = h / (a ​​/ 2) or applies H= (a / 2) cot10 ° = (a / 2) (cos10 ° / sin10 °)


First and second diagonal
......
The following applies sin40 ° = (i.e.1/ 2) / R or d1= 2Rsin40 ° = (sin40 °) / (sin20 °) a

We have sin60 ° = (i.e.2/ 2) / R or d2= 2Rsin60 ° = (sin60 °) / (sin20 °) a

R is replaced by R = a / (2sin20 °).


Third diagonal
......
The following applies sin80 ° = (i.e.3/ 2) / R or d3= 2Rsin80 ° = (sin80 °) / (sin20 °) a

R is replaced by R = a / (2sin20 °).


Area and perimeter
A.= 9 * [(1/2) ar] = 9 * [(1/2) a (a / 2) cot20 °] = (9/4) a² (cos20 ° / sin20 °)
U= 9a

Diagonals Top
All diagonals

The nine-corner has 27 diagonals.

......Nine diagonals connect every second, nine every third and nine every fourth corner point. The diagonals form three independent stars, the nonagrams.
 
Two stars can be drawn in one go. The middle star consists of three equilateral triangles. It arises from a triangle if you turn it 40 ° and 80 °.
The angles at the tips of the stars are 100 °, 60 ° and 20 °.

Another figure made of diagonals

endless

Squares and triangles

Enneagram
......The figure on the left consists of diagonals of the nine-corner.
I was astonished: She is probably the most frequently represented figure on the Internet in connection with the regular nine-corner.
You can find more information on Google with the keyword Enneagram or Enneagram, for example on Wikipedia (URL below). You leave the math.

On my Geoboard page you can find an overview of all convex figures that can form the diagonals.


Relationships between the diagonals Top
Various isosceles trapezoids lie in the triangle.

According to Ptolemy's theorem, there are relationships between the side and the diagonals:
(1) d3² = ad1+ d2² (2) d2² = ad3+ d1² (3) d1² = ad2+ a² (4) d3² = d2d3+ a² (5) d2² = ad3+ d1² (6) d3² = d1d2+ d1²

Ptolemy's theorem is explained and proven on the quadrilateral page.


......According to Ptolemy's theorem, one can deduce a simple relationship between two diagonals and the side:
d2d3= ad2+ d1d2 or d3= a + d1

This sentence is also more vivid.
......Because of the symmetry of the hexagon, the red diagonals are parallel. They form a yellow parallelogram with the sides d1 and a. Since the angles in the green triangle are 60 °, it is equilateral and side a also appears next to d1 on. So it's d1+ a = d3.

Draw a nine-corner Top
Not constructible

......It is well known that dividing an angle into three is generally not possible with a compass and ruler.
This also applies to the angle of 60 °.
The third part of an angle of 60 °, that is 20 °, cannot be constructed either.
Then an angle of 40 ° and thus the triangle cannot be constructed either.

drawing
......(1) Draw an angle of 40 °.
(2) Draw any circle around the vertex.
(3) Draw the string.
(4) Wear them on the arc of a circle.
Draw all the tendons.
The Neuneck was created.

Approximation construction
An approximate construction can be found in the same way as with the heptagon.
......> Draw an angle of 40 ° on squared paper.
> You can see that the free leg intersects a corner of the box at [7 to the right, 6 to the top]. This is used to approximate the angle of 40 °.
> The further construction is described above.
The equation tan (alpha) = 6/7 leads to an angle of 40.60 °. This angle is only 1.50% larger than 40 °.

"Neusis construction"
There is no construction of the three-way division of the angle of 60 °. If you allow a strip of paper with the possibility of marking a route, an exact drawing results, which is explained on my tripartite page.

1 Draw the given angle 3 * alpha = 60 ° and a circle around the vertex with an arbitrary radius r.
2 Mark a stretch of length r on a strip of paper and fit r between the horizontal and the circle on the outside so that the edge of the strip also runs through B.
3 The angle MDB is the desired angle alpha / 3 = 20 °.
Double the angle to get 40 degrees.

Regular neunagon on the Internet Top

German

Udo Hebisch (math café)
Nonagonreport, regular n-corner at Albrecht Dürer

Wikipedia
Neuneck, Enneagram, Palmanova



English

Dr Math (The Math Forum)
Nonagon or Enneagon?

Eric W. Weisstein
Nonagon, Nonagram, Trigonometry Angles - Pi / 9, Star of Goliath

John Fudjack
Taking the Mandala Literally

John Page
Nonagon

Nick Hobson
Puzzle 91. Nonagon diagonals, Solution
(Another proof by d3= a + d1 )

Wikipedia
Nonagon, Enneagon, Enneagram, Nonagonal number



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URL of my homepage:
http://www.mathematische-basteleien.de/

© 2005 Jürgen Köller

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