PENTAGON

{| border="1" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular pentagon
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|align=center colspan=2|
A regular pentagon, {5}
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|bgcolor=#e7dcc3|Edges and vertices||5
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|bgcolor=#e7dcc3|Schläfli symbol||{5}
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|bgcolor=#e7dcc3|Coxeter–Dynkin diagram||
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|bgcolor=#e7dcc3|Symmetry group||Dihedral (D₅)
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|bgcolor=#e7dcc3|Area
(with ''t''=edge length)||

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|bgcolor=#e7dcc3|Internal angle
(degrees)||108°
|}
In geometry, a 'pentagon' is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540°.

Contents

Regular pentagons

Construction

See also

External links

Regular pentagons

The term ''pentagon'' is commonly used to mean a 'regular convex pentagon', where all sides are equal and all interior angles are equal (to 108°). Its Schläfli symbol is {5}.
The area of a regular convex pentagon with side length ''t'' is given by

A 'pentagram' is a 'regular star pentagon'. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon - in this arrangement the sides of the two pentagons are in the golden ratio.

Construction

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



#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).
#Choose a point ''A'' on the circle that will serve as one vertex of the pentagon. Draw a line through ''O'' and ''A''.
#Construct a line perpendicular to the line ''OA'' passing through ''O''. Mark its intersection with one side of the circle as the point ''B''.
#Construct the point ''C'' as the midpoint of ''O'' and ''B''.
#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''.
#Draw a circle centered at ''A'' through the point ''D''. Mark its intersections with the original (green) circle as the points ''E'' and ''F''.
#Draw a circle centered at ''E'' through the point ''A''. Mark its other intersection with the original circle as the point ''G''.
#Draw a circle centered at ''F'' through the point ''A''. Mark its other intersection with the original circle as the point ''H''.
#Construct the regular pentagon ''AEGHF''.
After forming a regular convex pentagon, if 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.
An alternative method of construction is illustrated in the animation:

See also

★ Trigonometric constants for a pentagon

★ Pentagram

External links

★

★ How to construct a regular pentagon using only compass and straightedge

★ Definition and properties of the pentagon, with interactive animation

★ Nine constructions for the regular pentagon by Robin Hu

★ Renaissance artists' approximate constructions of regular pentagons at Convergence