During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.

Difference between revisions of "Pentagon"

(Removed irrelevant paragraph on the occult properties of the pentagram)
Line 1: Line 1:
In [[geometry]], a '''pentagon''' is a [[polygon]] with 5 sides. The sum of the internal angles of any pentagon is <math>540^{\circ}</math>. Each [[angle]] of a [[regular polygon | regular]] pentagon is <math>108^{\circ}</math>.  
+
In [[geometry]], a '''pentagon''' is a [[polygon]] with 5 sides. Each [[angle]] of a [[regular polygon | regular]] pentagon is <math>108^{\circ}</math>. The sum of the internal angles of any pentagon is <math>540^{\circ}</math>.
  
 
== Construction ==
 
== Construction ==
Line 15: Line 15:
  
 
==The Golden Ratio and the Pentagram==
 
==The Golden Ratio and the Pentagram==
The regular 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>.
+
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>.
+
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 ==

Revision as of 19:38, 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

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

Invalid username
Login to AoPS