Octagon Definition - Math Open Reference

Octagon (8-gon)

From Greek: okto "eight" + gonia "angle"

Definition: A polygon with 8 sides

Try this Adjust the octagon below by dragging any orange dot. By clicking on the top left command line, you can switch it between a regular and irregular octagon.

(If there is no image below, see support page.)

Because an octagon has an even number of sides, in a regular octagon, opposite sides are parallel.

Properties of regular octagons

Interior angle	135°	Like any regular polygon, to find the interior angle we use the formula (180n–360)/n . For an octagon, n=8. See Interior Angles of a Polygon
Exterior Angle	45°	To find the exterior angle of a regular octagon, we use the fact that the exterior angle forms a linear pair with the interior angle, so in general it is given by the formula 180-interior angle. See Exterior Angles of a Polygon
Area	4.828s² approx	Where S is the length of a side. To find the exact area of an octagon or any polygon, using various methods, see Area of a Regular Polygon and Area of an Irregular Polygon

Properties of all octagons

Number of diagonals	20	The number of distinct diagonals possible from all vertices. (In general ½n(n–3) ). In the figure above, click on "show diagonals" to see them. See Diagonals of a Polygon
Number of triangles	6	The number of triangles created by drawing the diagonals from a given vertex. (In general n–2). In the figure above, click on "show triangles" to see them. See Triangles of a Polygon
Sum of interior angles	1080°	In general 180(n–2) degrees . See Interior Angles of a Polygon

The Octagon Houses

Houses and rooms are sometimes built in an octagonal shape (pronounced: "ock-TAG-on-all"), perhaps because they feel somewhat like a circular space but are made of straight wall sections which are easier to build.

While no longer so popular, there are examples that are being preserved as historic buildings. The one on the right is the Loren Andrus Octagon House in Washington, Michigan USA, built in 1860.

Because they are based on a regular octagon, opposite walls are parallel and the rooms have a pleasing symmetry.

Related polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons