Eighteenth

regular eighteenth

An eighteenth or octodecagon is a geometric figure . It belongs to the group of polygons ( polygons ). It is defined by eighteen corner points .

Regular eighteenth

A special case of the eighteenth is the regular eighteenth whose sides are all of the same length and whose corner points lie on a common circumference .

Mathematical relationships

Formula for angle calculations

The angle ( central angle ) enclosed by two circumferential radii passing through adjacent corners is

${\displaystyle \alpha ={\frac {360^{\circ }}{18}}=20^{\circ }}$.

The angle that two adjacent side edges enclose with one another in the flat, regular eighteenth corner is

${\displaystyle \beta ={\frac {(18-2)}{18}}\cdot 180^{\circ }={\frac {16}{18}}\cdot 180^{\circ }=160^{\circ }}$.

Formula for the side length

${\displaystyle s=2\cdot r_{u}\cdot \sin {\frac {\alpha }{2}}}$

${\displaystyle s=2\cdot r_{u}\cdot \sin {10^{\circ }}}$

${\displaystyle s\approx r_{u}\cdot 0{,}1736481777}$

Diagonals

Eighteen-corner with diagonals

In the regular eighteenth corner there are a total of 135 diagonals with different lengths:

• 18 diagonals over 2 (or 16) sides
• 18 diagonals over 3 (or 15) sides
• 18 diagonals over 4 (or 14) sides
• 18 diagonals over 5 (or 13) sides
• 18 diagonals over 6 (or 12) sides
• 18 diagonals over 7 (or 11) sides
• 18 diagonals over 8 (or 10) sides
• 9 diagonals over 9 sides

The lengths in relation to the perimeter radius are:

• The diagonal across two sides corresponds to the side of a hexagon with the same circumference:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {20^{\circ }}\approx r_{u}\cdot 0{,}68404029}$
• The diagonal across three sides corresponds to the side of a hexagon with the same circumference:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {30^{\circ }}=r_{u}}$
• The diagonal over four sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {40^{\circ }}\approx r_{u}\cdot 1{,}28557522}$
• The diagonal over five sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {50^{\circ }}\approx r_{u}\cdot 1{,}53208889}$
• The diagonal across six sides corresponds to the side of an equilateral triangle with the same circumference:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {60^{\circ }}=r_{u}\cdot {\sqrt {3}}}$
• The diagonal over seven sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {70^{\circ }}\approx r_{u}\cdot 1{,}87938524}$
• The diagonal over eight sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {80^{\circ }}\approx r_{u}\cdot 1{,}96961551}$
• The diagonal over nine sides corresponds to the diameter of the circumference:
  • ${\displaystyle s=2\cdot r_{u}}$

Formula for the area

Construction of the eighteenth, a modification of the construction of the nine with tomahawk .

The area A of the regular eighteenth corner with side length s is

${\displaystyle A={\frac {18}{4}}\cdot s^{2}\cdot {\frac {\cos 10^{\circ }}{\sin 10^{\circ }}}}$

or with the perimeter radius

${\displaystyle A={\frac {18}{2}}\cdot r_{u}^{2}\cdot \sin 20^{\circ }}$.

construction

The eighteenth is constructed by constructing a hexagon (for details see article there) and then bisecting the angle or side. Because a regular nine-corner cannot only be constructed with a compass and ruler , the same applies to the eighteenth-corner. You need at least one tool to divide an angle into three .

Regular eighteen corners

A regular overturned eighteenth arises if at least one is skipped over each time when connecting the eight corner points and the chords thus created are of the same length. Such regular stars are noted with Schläfli symbols , indicating the number of corner points and connecting every -th point. ${\displaystyle \left\{n/k\right\}}$ ${\displaystyle n}$ ${\displaystyle k}$

There are only two regular eighteenth- pointed stars , also called octodecagrams .

The "stars" with the Schläfli symbols {18/2} and {18/16} are triangles , those with the Schläfli symbols {18/3} and {18/15} are hexagons and those with the Schläfli symbols { 18/6} and {18/12} are equilateral triangles . The stars with the Schläfli symbols {18/4} and {18/14} or {18/8} and {18/10} are regular nine-ray stars, also called enneagrams.

• Regular eighteen-ray stars
• 

  ${\displaystyle \left\{18/5\right\}{,}\ \left\{18/13\right\}}$

• 

  ${\displaystyle \left\{18/7\right\}{,}\ \left\{18/11\right\}}$

Use of the eighteenth

Liberation Hall

The Liberation Hall in Kelheim , Bavaria, built around 1850 to commemorate victories over Napoleon, has an eighteen-cornered floor plan.