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 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

Derivation of the Area formula

The area of any regular polygon is:

where P is the perimeter of the polygon, and a is the apothem. We can then substitute the respective values for P and a, which makes the formula:

with t as the given side length. Then we can then rearrange the formula as:

and then, we combine the two terms to get the final formula, which is:

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