Pentagon

Regular pentagon
A regular pentagon, {5}
Edges and vertices	5
Schläfli symbol	{5}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D5)
Area (with t=edge length)	$\backslash frac\{\{t^2\; \backslash sqrt\; \{25\; +\; 10\backslash sqrt\; 5\; \}\; \}\}\{4\}$ $\backslash approx\; 1.720477401\; t^2.$
Internal angle (degrees)	108°

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This article is about the geometric figure. For the headquarters of the United States Department of Defense, see The Pentagon. For other uses, see Pentagon (disambiguation).

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°.

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 $A\; =\; \backslash frac\{\{t^2\; \backslash sqrt\; \{25\; +\; 10\backslash sqrt\; 5\; \}\; \}\}\{4\}\; =\; \backslash frac\{5t^2\; \backslash cdot\; \backslash tan(54^\backslash circ)\}\{4\}\backslash \; \backslash approx\; 1.720477401\; t^2.$

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:

Construction of a regular pentagon

An alternative method is this:

Constructing a pentagon

1. 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).
2. Choose a point A on the circle that will serve as one vertex of the pentagon. Draw a line through O and A.
3. Construct a line perpendicular to the line OA passing through O. Mark its intersection with one side of the circle as the point B.
4. Construct the point C as the midpoint of O and B.
5. 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.
6. Draw a circle centered at A through the point D. Mark its intersections with the original (green) circle as the points E and F.
7. Draw a circle centered at E through the point A. Mark its other intersection with the original circle as the point G.
8. Draw a circle centered at F through the point A. Mark its other intersection with the original circle as the point H.
9. 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.

Pentagons in nature

See also

• Trigonometric constants for a pentagon
• Pentagram
• The Pentagon
• Pentastar
• Dodecahedron, a polyhedron whose regular form is composed of 12 pentagonal faces

External links

• Eric W. Weisstein, Pentagon at MathWorld.
• 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

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