# Difference between revisions of "Pentagon"

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== Construction == | == Construction == | ||

− | + | [[Image:Pentagon.png|center]] | |

It is possible to construct a regular pentagon with compass and straightedge: | It is possible to construct a regular pentagon with compass and straightedge: | ||

− | + | # Draw circle <math>O</math> (red). | |

+ | # Draw diameter <math>AB</math> and construct a perpendicular radius through <math>O</math>. | ||

+ | # Construct the midpoint of <math>CO</math>, and label it <math>E</math>. | ||

+ | # Draw <math>AE</math> (green). | ||

+ | # Construct the angle bisector of <math>\angle AEO</math>, and label its intersection with <math>AB</math> as <math>F</math> (pink). | ||

+ | # Construct a perpendicular to <math>AB</math> at <math>F</math>. | ||

+ | # Adjust your compass to length <math>AG</math>, and mark off points <math>H</math>, <math>I</math> and <math>J</math> on circle <math>O</math>. | ||

+ | # <math>AGHIJ</math> is a regular pentagon. | ||

− | 1. | + | ==The Golden Ratio and the Pentagram== |

+ | The pentagon is closely associated with the [[Golden Ratio]]. More specifically, the ratio of a diagonal to an edge is <math>\frac{1+\sqrt{5}}{2}</math>. By drawing each of the diagonals, one can form a pentagram, or five-pointed star, in which each of the internal angles is <math>36^{\circ}</math>.\\ | ||

− | + | == See Also == | |

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− | == See | ||

*[[Polygon]] | *[[Polygon]] | ||

{{stub}} | {{stub}} | ||

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+ | [[Category:Definition]] | ||

+ | [[Category:Geometry]] |

## Latest revision as of 19:39, 20 July 2016

In geometry, a **pentagon** is a polygon with 5 sides. Each angle of a regular pentagon is . The sum of the internal angles of any pentagon is .

## Construction

It is possible to construct a regular pentagon with compass and straightedge:

- Draw circle (red).
- Draw diameter and construct a perpendicular radius through .
- Construct the midpoint of , and label it .
- Draw (green).
- Construct the angle bisector of , and label its intersection with as (pink).
- Construct a perpendicular to at .
- Adjust your compass to length , and mark off points , and on circle .
- is a regular pentagon.

## The Golden Ratio and the Pentagram

The pentagon is closely associated with the Golden Ratio. More specifically, the ratio of a diagonal to an edge is . By drawing each of the diagonals, one can form a pentagram, or five-pointed star, in which each of the internal angles is .\\

## See Also

*This article is a stub. Help us out by expanding it.*