Pentagon

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For other uses, see Pentagon (disambiguation).
For the headquarters of the United States Department of Defense, see The Pentagon.
A regular pentagon
A regular 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 = \frac{5a^2}{4}\cot \frac{\pi}{5} = \frac {a^2}{4} \sqrt{25+10\sqrt{5}} \approx 1.72048 a^2

[edit] Constructing a pentagon

Construction of a regular pentagon
Construction of a regular pentagon

A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC.

One method to construct a regular pentagon in a given circle is as follows:

Constructing a pentagon

  1. Draw a circle in which to inscribe the pentagon and mark the center point O. (This is the green circle in the diagram to the right).
  2. Choose a point A on the circle that will serve as one vertex of the pentagon. Draw a line through O and A.
  3. Construct a line perpendicular to the line OA passing through O. Mark its intersection with one side of the circle as the point B.
  4. Construct the point C as the midpoint of O and B.
  5. Draw a circle centered at C through the point A. Mark its intersection with the line OB (inside the original circle) as the point D.
  6. Draw a circle centered at A through the point D. Mark its intersections with the original (green) circle as the points E and F.
  7. Draw a circle centered at E through the point A. Mark its other intersection with the original circle as the point G.
  8. Draw a circle centered at F through the point A. Mark its other intersection with the original circle as the point H.
  9. Construct the regular pentagon AEGHF.

After forming 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.

An alternative method of construction is illustrated in the animation: Constructing a regular pentagon with compass and straightedge.

[edit] Some relevant trigonometric values

\sin \frac{\pi}{10} = \sin 18^\circ = \frac{\sqrt 5 - 1}{4}
\cos \frac{\pi}{10} = \cos 18^\circ = \frac{\sqrt{2(5 + \sqrt 5)}}{4}
\tan \frac{\pi}{10} = \tan 18^\circ = \frac{\sqrt{5(5 - 2 \sqrt 5)}}{5}
\cot \frac{\pi}{10} = \cot 18^\circ = \sqrt{5 + 2 \sqrt 5}
\sin \frac{\pi}{5} = \sin 36^\circ = \frac{\sqrt{2(5 - \sqrt 5)} }{4}
\cos \frac{\pi}{5} = \cos 36^\circ = \frac{\sqrt 5+1}{4}
\tan \frac{\pi}{5} = \tan 36^\circ = \sqrt{5 - 2\sqrt 5}
\cot \frac{\pi}{5} = \cot 36^\circ = \frac{\sqrt{5(5 + 2\sqrt 5)}}{5}

[edit] External links


Polygons
Triangle • Quadrilateral • Pentagon •Hexagon • Heptagon • Octagon • Enneagon (Nonagon) • Decagon • Hendecagon • Dodecagon • Triskaidecagon • Pentadecagon • Hexadecagon • Heptadecagon • Enneadecagon • Icosagon • Chiliagon • Myriagon
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