Pentagon

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Image:Regular pentagon.png In geometry, a pentagon is any five-sided polygon. However, the term is commonly used to mean a regular pentagon, where all sides are equal and all angles are equal (to 108°). Its Schläfli symbol is {5}.

The area of a regular pentagon with side length a is given by <math>A = \frac{5a^2}{4}\cot \frac{\pi}{5} = \frac {a^2}{4} \sqrt{25+10\sqrt{5}} \simeq 1.72048 a^2</math>
Image:Regular pentagram.png A pentagram can be formed from a regular pentagon either by extending its sides or by drawing its diagonals. The two differ by a linear scale factor φ + 1, or conversely 2 - φ, where φ = (1+√5)/2, the golden ratio. The resulting figure contains also many more various other lengths related by the golden ratio.

Constructing a pentagon

1. Put the needle in (b) and pass a circle segment through (c) and the first circle. These points on the first circle are the second and third corners of the pentagon.
2. Without extending the compass, put its needle in the second and third corners, and draw circle segments passing through the first circle to find the two remaining corners.
3. Join each corner to the adjacent ones and you have a pentagon.
4. If you join the non-adjacent corners (drawing the diagonals of the pentagon), you obtain a pentagram, with a smaller regular pentagon in the center. Or if you extend the sides until the non-adjacent ones meet, you obtain a larger pentagram.

Image:Pentagon construct.gif

Some relevant trigonometric values

<math>\sin \frac{\pi}{10} = \sin 18^\circ = \frac{\sqrt 5 - 1}{4}</math>

<math>\cos \frac{\pi}{10} = \cos 18^\circ = \frac{\sqrt{2(5 + \sqrt 5)}}{4} </math>

<math>\tan \frac{\pi}{10} = \tan 18^\circ = \frac{\sqrt{5(5 - 2 \sqrt 5)}}{5} </math>

<math>\cot \frac{\pi}{10} = \cot 18^\circ = \sqrt{5 + 2 \sqrt 5} </math>

<math>\sin \frac{\pi}{5} = \sin 36^\circ = \frac{\sqrt{2(5 - \sqrt 5)} }{4}</math>

<math>\cos \frac{\pi}{5} = \cos 36^\circ = \frac{\sqrt 5+1}{4}</math>

<math>\tan \frac{\pi}{5} = \tan 36^\circ = \sqrt{5 - 2\sqrt 5} </math>

<math>\cot \frac{\pi}{5} = \cot 36^\circ = \frac{ \sqrt{5(5 + 2\sqrt 5)}}{5} </math>

External links

• Pentagons & Pentagrams new facts about pentagons and pentagrams by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas. Key concept: Menelaus Theorem.

• Quasi crystal inspired pentagon tile by Alexander Braun.

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