# Pentagon

## Pentagon

Polygon with five sides.

### Properties

• A pentagon is called a regular pentagon if all its sides are congruent and if all its angles are congruent.
• A pentagon has five interior angles. If it is a regular pentagon, each of its interior angles measures 108°.
• A pentagon has 5 vertices, 5 sides and 5 diagonals.

### Formulas

The formula to calculate the area A of a regular pentagon of side length c is:

$$A=\frac{c^{2}}{4}\sqrt{\left ( 25+10\sqrt{5} \right )}$$

The formula to calculate the radius r of the circle circumscribed about a regular pentagon of side length c is:

$$r=\frac{c}{10}\sqrt{\left ( 50+10\sqrt{5} \right )}$$

The formula to calculate the radius r of the circle inscribed in a regular pentagon of side length c is:

$$r=\frac{c}{10}\sqrt{\left ( 25+10\sqrt{5} \right )}$$

### Example

This is a regular pentagon inscribed in a circle:

### Ethymological notes

• The term “pentagon” is derived from the Latin pentagonum the nominalization of the adjective pentagonus, which was borrowed from the ancient Greek πεντάγωνος (pentágônos), “pentagonal”, “which has five angles, five sides”. The Greek term itself is composed of the terms πέντε (pénte), “five,” and γωνία (gônía), “angle”.
• The Greek term appears in Book IV of Euclid’s Elements, which was likely written around 300 BCE and deals with inscribed and circumscribed figures, in particular regular polygons.