Octagon

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Octagon

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an octagon is a polygon

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
that has eight

8 (number)

8 is the natural number, following 7 and preceding 9 . The SI prefix for 10008 is yotta , and for its reciprocal yocto . It is the root of two other numbers: eighteen and eighty ....
sides. A regular octagon is represented by the Schläfli symbol

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
.
gular octagon is always an octagon whose sides are all the same length and whose internal angles are all the same size. The internal angle

Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
at each vertex of a [ irregular and regular

[regular polygon|regular]] octagon is 135°

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
and the sum of all the internal angles is 1080°

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
. The area of a regular octagon of side length a is given by

In terms of , (circumradius) the area is

In terms of , (inradius

Inscribe

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. Specifically, at all points where figures meet, their edges must lie tangent....
) the area is

Naturally, those last two coefficients bracket the value of pi

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
, the area of the unit circle

Unit circle

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Regular octagons

A regular octagon is always an octagon whose sides are all the same length and whose internal angles are all the same size. The internal angle

Angle

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
and the sum of all the internal angles is 1080°

Degree (angle)

In terms of , (circumradius) the area is

In terms of , (inradius

Inscribe

Naturally, those last two coefficients bracket the value of pi

Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
.

The area may also be found this way: Where is the span of the octagon, or the second shortest diagonal; and is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45-45-90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.

Given the span , the length of a side is

The area, then, is

Uses of octagons

In many parts of the world, stop sign Stop sign A stop sign is a traffic sign, usually erected at road junctions, that instructs drivers to stop and then to proceed only if the way ahead is clear.... s are in the shape of a regular octagon.	Push-button

An eight-sided star Star polygon A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined.... , called an octagram, with Schläfli symbol is contained with a regular octagon.	The vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.... of the uniform polyhedron Uniform polyhedron A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry.... , great dirhombicosidodecahedron Great dirhombicosidodecahedron In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex.... is contained within an irregular Irregular Something that is irregular does not follow the expected pattern. The term is used in many different fields, with quite different meanings.* In astronomy, an astronomical object whose shape or behavior varies considerably from the archetype is called an irregular.... 8-sided star polygon, with four edges going through its center.	An octagonal prism Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron.... contains two octagons.
The truncated square tiling Truncated square tiling In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon.... has 2 octagons around every vertex.	The truncated cuboctahedron Truncated cuboctahedron The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges.... has 6 octagons	An octagonal antiprism Octagonal antiprism In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.... contains two octagons.	The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.

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With interactive animation

Octagon

Regular octagons

Uses of octagons

See also

External links