eightsidedness

Eightsidedness is the property of having eight sides in a polygon. In mathematical writing, the standard term is octagon or octagonality; eightsidedness is a descriptive, informal way to refer to that characteristic.

An eight-sided polygon is called an octagon. If the octagon is regular, all its sides and all its interior angles are equal. The sum of the interior angles of any octagon is (n−2)×180 degrees, where n is 8, giving 1080 degrees. Therefore, in a regular octagon each interior angle measures 135 degrees, and each exterior angle measures 45 degrees. The number of diagonals in an octagon is n(n−3)/2, which equals 20.

Octagons exhibit dihedral symmetry, with eight rotational symmetries and eight reflections, forming the dihedral group D8. This symmetry underpins many geometric constructions and design patterns.

In tiling and design, eight-sided figures occur in various contexts. Regular octagons do not tile the plane by themselves, but they can participate in semi-regular tilings, such as those combining octagons with squares. In practical use, the most familiar example of an eight-sided figure is the stop sign.

Variations of eightsidedness include convex and concave (non-convex) eight-sided polygons, as well as irregular octagons with unequal side lengths and angles. Regardless of regularity, the interior angle sum remains 1080 degrees.

Etymology: octa- means eight and gonia means angle, from Greek; the noun form octagon describes the eight-sided polygon, while eightsidedness denotes the quality itself.