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.
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
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 °)
|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 °).
|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 °)
|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
Squares and triangles
|......||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) 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:|
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
|......||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.
|......||(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.
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.
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
Udo Hebisch (math café)
Nonagonreport, regular n-corner at Albrecht Dürer
Neuneck, Enneagram, Palmanova
Dr Math (The Math Forum)
Nonagon or Enneagon?
Eric W. Weisstein
Nonagon, Nonagram, Trigonometry Angles - Pi / 9, Star of Goliath
Taking the Mandala Literally
Puzzle 91. Nonagon diagonals, Solution
(Another proof by d3= a + d1 )
Nonagon, Enneagon, Enneagram, Nonagonal number
Feedback: Email address on my main page
URL of my homepage:
© 2005 Jürgen Köller
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