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Difference between revisions of "Pentagon"

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# 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>.
 
# 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.
 
# <math>AGHIJ</math> is a regular pentagon.
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==The Golden Ratio and the Pentagram==
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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 ==
 
== See Also ==

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 $108^{\circ}$. The sum of the internal angles of any pentagon is $540^{\circ}$.

Construction

Pentagon.png

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

  1. Draw circle $O$ (red).
  2. Draw diameter $AB$ and construct a perpendicular radius through $O$.
  3. Construct the midpoint of $CO$, and label it $E$.
  4. Draw $AE$ (green).
  5. Construct the angle bisector of $\angle AEO$, and label its intersection with $AB$ as $F$ (pink).
  6. Construct a perpendicular to $AB$ at $F$.
  7. Adjust your compass to length $AG$, and mark off points $H$, $I$ and $J$ on circle $O$.
  8. $AGHIJ$ 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 $\frac{1+\sqrt{5}}{2}$. By drawing each of the diagonals, one can form a pentagram, or five-pointed star, in which each of the internal angles is $36^{\circ}$.\\

See Also

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