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Encyclopedia > Pentagon
Look up pentagon in Wiktionary, the free dictionary.
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}. Wikipedia does not have an article with this exact name. ... Wiktionary is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... Pentagon can refer to: Pentagon - a five-sided polygon Pentagon - a Soviet clone of the ZX Spectrum computer The Pentagon - the headquarters of the United States Department of Defense, in Arlington County, Virginia Pentagon (chess) - a chess engine Tritec Pentagon engine for automobiles This is a disambiguation page — a... Image File history File links Pentagon. ... Image File history File links Pentagon. ... Table of Geometry, from the 1728 Cyclopaedia. ... Look up Polygon in Wiktionary, the free dictionary. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...

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+10sqrt{5}} approx 1.72048 a^2

Constructing a 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. In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Euclid (also referred to as Euclid of Alexandria) (Greek: ) (c. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...

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

Constructing a pentagon Image File history File links Pentagon-construction. ...

  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.

If, and so you have formed a pentagon 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. Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... A pentagram, pentalpha, or pentangle A pentagram (sometimes known as pentalpha or pentangle) is a five-pointed star drawn with five straight strokes. ...

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

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 - 2sqrt 5}
cot frac{pi}{5} = cot 36^circ = frac{sqrt{5(5 + 2sqrt 5)}}{5}

External links

Triangle | Quadrilateral | Pentagon | Hexagon | Heptagon | Octagon | Enneagon (Nonagon) | Decagon | Hendecagon | Dodecagon | Triskaidecagon | Pentadecagon | Hexadecagon | Heptadecagon | Enneadecagon | Icosagon | Icosihenagon | Tricontagon | Pentacontagon | Hectagon | Chiliagon | Myriagon

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That side of the Pentagon was virtually empty and had been for some time because the building had been under reconstruction quite visibly on that side for 5-6 years and part of that was to reinforce the building from external attack.
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