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Pentagon




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
A = rac{5a^2}{4}\cot rac{\pi}{5} = rac {a^2}{4} \sqrt{25+10\sqrt{5}} \approx 1.72048 a^2



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


#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.
#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.
#Join each corner to the adjacent ones and you have a pentagon.
#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.


SOME RELEVANT TRIGONOMETRIC VALUES

:\sin rac{\pi}{10} = \sin 18^\circ = rac{\sqrt 5 - 1}{4}
:\cos rac{\pi}{10} = \cos 18^\circ = rac{\sqrt{2(5 + \sqrt 5)}}{4}
: an rac{\pi}{10} = an 18^\circ = rac{\sqrt{5(5 - 2 \sqrt 5)}}{5}
:\cot rac{\pi}{10} = \cot 18^\circ = \sqrt{5 + 2 \sqrt 5}

:\sin rac{\pi}{5} = \sin 36^\circ = rac{\sqrt{2(5 - \sqrt 5)} }{4}
:\cos rac{\pi}{5} = \cos 36^\circ = rac{\sqrt 5+1}{4}
: an rac{\pi}{5} = an 36^\circ = \sqrt{5 - 2\sqrt 5}
:\cot rac{\pi}{5} = \cot 36^\circ = rac{ \sqrt{5(5 + 2\sqrt 5)}}{5}


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.