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A regular pentagon, {5}

Regular pentagon
Edges and vertices 5
Schläfli symbol {5}
Coxeter–Dynkin diagram $\frac{{t^2 \sqrt {25 + 10\sqrt 5 } }}{4}$
Symmetry group Dihedral (D5)
Area
(with t=edge length)
$\frac{{t^2 \sqrt {25 + 10\sqrt 5 } }}{4}$
$\frac{{t^2 \sqrt {25 + 10\sqrt 5 } }}{4}$
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°.

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:

Construction of a regular 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.

Constructing a pentagon

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.