# Pentagon

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Regular pentagon

A regular pentagon, {5}
Edges and vertices5
Schläfli symbol{5}
Coxeter–Dynkin diagram
Symmetry groupDihedral (D5)
Area
(with $a$ = edge length)
$\frac{\sqrt{25+10\sqrt5}}{4}a^2$
$\approx1.72a^2$
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 $540^\circ$ .

## 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 $a$ is given by

$A=\frac{\sqrt{25+10\sqrt5}}{4}a^2=\frac{5\tan(54^\circ)}{4}a^2\approx1.72a^2$

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 $a$ is given by the expression

$a=\sqrt{\frac{5-\sqrt5}{2}}R=2\sin(36^\circ)R=2\sin\left(\frac{\pi}{5}\right)R\approx1.175R$

### 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 $OB$ .
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.

A simple method of creating a regular pentagon from just a strip of paper is by tying an overhand knot into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon will reveal a pentagram when backlit.

## Graphs

The $K_5$ complete graph is often drawn as a regular pentagon with all 10 edges connected. This graph also represents an orthographic projection of the 5 vertices and 10 edges of the 5-cell. The rectified 5-cell, with vertices at the mid-edges of the 5-cell is projected inside a pentagon.

 5-cell (4D) Rectified 5-cell (4D)