|- |bgcolor=#e7dcc3|Edges and vertices||5 |- |bgcolor=#e7dcc3|Schläfli symbol||{5} |- |bgcolor=#e7dcc3|Coxeter–Dynkin diagram||

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 chords of this pentagon are in golden ratio to its sides.

The area of a regular convex pentagon with side length t is given by

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

A pentagram or pentangle 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.

When a regular pentagon is inscribed in a circle with radius R, its edge length t is given by the expression

$t\; =\; R\backslash \; \{\backslash sqrt\; \{\; \backslash frac\; \{5-\backslash sqrt\{5\}\}\{2\}\}\; \}\; =\; 2R\backslash sin\; 36^\backslash circ\; =\; 2R\backslash sin\backslash frac\{\backslash pi\}\{5\}\; \backslash approx\; 1.17557050458\; R.$

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:

An alternative method is this:

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.