Octagon

A generalization van Aubel theorem

Octagon in square

8-folding

Regular Octagon Inscribed in a Circle

Meccano octagon

8-cube t0 A7

Octagon is a geometric shape that features eight straight sides and eight angles. The term originates from the Greek word oktagōnos, which means "eight angles". Octagons are commonly found in various aspects of architecture, design, and nature. They can be regular or irregular, with regular octagons having all sides of equal length and all internal angles equal to 135 degrees.

Characteristics

A regular octagon is a closed figure in geometry with sides of the same length and internal angles of the same size. It can be constructed as a polygon with eight sides and fits into the broader category of n-gons, where n represents the number of sides.

Area and Perimeter

The area A of a regular octagon can be calculated using the formula: \[A = 2(1 + \sqrt{2})s^2\] where s is the length of a side. The perimeter P of a regular octagon is given by: \[P = 8s\] where s is again the length of a side.

Applications

Octagons are prevalent in various fields, including architecture, urban planning, and design. One of the most recognized uses of the octagon shape is in the design of stop signs used in road traffic control, which are regular octagons with a red background and white lettering in many countries. This shape is chosen for its unique properties, making it easily recognizable and giving it the ability to stand out among other road signs.

In architecture, octagonal patterns are often used in the design of buildings, tiles, and decorative elements. The octagon is also significant in religious architecture, such as in Islamic and Christian structures, where it may symbolize regeneration or infinity.

Mathematical Properties

In mathematics, the octagon is studied for its properties and its relationship with other geometric figures. For example, a regular octagon can be constructed by connecting the vertices of a square at the midpoints of its sides and then cutting off the corners to form an octagon. This demonstrates the close relationship between octagons and other polygons.

Variations

Besides the regular octagon, there are various irregular octagonal shapes with sides and angles of different lengths and sizes. These can be convex or concave, depending on the arrangement of the vertices.

In Culture

The octagon also appears in cultural contexts, such as in the layout of certain historical buildings and in symbols. It can represent elements such as balance, renewal, and transition, depending on the cultural interpretation.

See Also

• Polygon
• Geometry
• Square
• Hexagon
• Stop sign