# Octagon

## Octagon

A polygon with eight sides.

### Properties

• An octagon is a regular octagon if all its sides are congruent and if all its interior angles have the same measure.
• An octagon has eight interior angles and, if it is a regular octagon, each of its interior angles measures 135°.
• The sum of the interior angles of a convex octagon is 1080°.

### Formula

Consider an octagon with a side length c and an apothem a in which r is the radius of the circumscribed circle.

A formula to calculate the area A of a regular octagon of radius $$r$$ is:

$$A=2r^{2}\sqrt{2}$$.

Another formula to calculate the area A of a regular octagon of side length c and apothem a is:

$$A=4ca$$.

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

$$A=2c^{2}\left ( 1+\sqrt{2}\right )$$.

### Exemple

This is a regular octagon inscribed in a circle that has an apothem a: