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. | |
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| − | + | ==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 | + | == See Also == |
*[[Polygon]] | *[[Polygon]] | ||
{{stub}} | {{stub}} | ||
| + | [[Category:Definition]] | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
Revision as of 20:39, 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.
(red).
and construct a perpendicular radius through 
, and label it 
.
(green).
, and label its intersection with 
(pink).
, 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
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