# What is a nonagon in math

**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. |

*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 **d _{1}, d_{2 }**and

**d**and the height

_{3}**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 |

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 |

First and second diagonal

The following applies sin40 ° = (i.e._{1}/ 2) / R or d= 2Rsin40 ° = (sin40 °) / (sin20 °) a_{1}We have sin60 ° = (i.e. R is replaced by R = a / (2sin20 °). |

Third diagonal

The following applies sin80 ° = (i.e._{3}/ 2) / R or d= 2Rsin80 ° = (sin80 °) / (sin20 °) a_{3}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. |

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. |

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.

(1) d

_{3}² = ad

_{1}+ d

_{2}² (2) d

_{2}² = ad

_{3}+ d

_{1}² (3) d

_{1}² = ad

_{2}+ a² (4) d

_{3}² = d

_{2}d

_{3}+ a² (5) d

_{2}² = ad

_{3}+ d

_{1}² (6) d

_{3}² = d

_{1}d

_{2}+ d

_{1}²

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:_{2}d_{3}= ad_{2}+ d_{1}d_{2} or d_{3}= a + d_{1} |

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 d_{1} and a. Since the angles in the green triangle are 60 °, it is equilateral and side a also appears next to d_{1} on. So it's d_{1}+ a = d_{3}. |

**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. |

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. |

"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.

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 d _{3}= a + d_{1} )*

Wikipedia

Nonagon, Enneagon, Enneagram, Nonagonal number

**Feedback:**Email address on my main page

URL of my homepage:

http://www.mathematische-basteleien.de/

© 2005 Jürgen Köller

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