Regular octagon
A regular octagon
Edges and vertices	8
Schläfli symbols	{8} t{4}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D8)
Area (with t=edge length)
Internal angle (degrees)	135°

In geometry, an octagon is a polygon that has eight sides. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In mathematics 8 is a Composite number, its Proper divisors being 1, 2, and 4. A regular octagon is represented by the Schläfli symbol {8}. In Mathematics, the Schläfli symbol is a notation of the form {pqr

Regular octagons

A regular octagon is constructible with compass and straightedge. In mathematics a constructible polygon is a Regular polygon that can be constructed with compass and straightedge. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles To do so, follow steps 1 through 18 of the animation, noting that the compass radius is not altered during steps 7 through 10.

A regular octagon is an octagon whose sides are all the same length and whose internal angles are all the same size. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080°. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called General properties These properties apply to both convex and star regular polygons This article describes the unit of angle For other meanings see Degree. This article describes the unit of angle For other meanings see Degree.

The area of a regular octagon of side length a is given by

In terms of R, (circumradius) the area is

In terms of r, (inradius) the area is

Naturally, those last two coefficients bracket the value of pi, the area of the unit circle. In Geometry, an inscribed Planar Shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid In Mathematics, a coefficient is a Constant multiplicative factor of a certain object IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems In Mathematics, a unit circle is
The area may also be found this way:

A = S² − B².

Where S is the span of the octagon, or the second shortest diagonal; and B 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. Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised The edges of this square are each the length of the base. This is the easier way to find such an area. However, one of the most accurate ways is as follows: Take the coefficient of the square root of one of the sides and multiply by three! This is proved by the fact that coefficient of pi, proved by Archidimes, is equal to two times the sin of any given triangle as long as all equal sides add up to the factorial of the scalene ones, especially those of such a square. Seeing as an octagon has three of these triangles in the interior of a convex Given the span S the length of a side B is

/ ,S1 + S2 + S3 + S4 + S5 + S6 +S7 + S8

Uses of octagons

In many parts of the world, stop signs are in the shape of a regular octagon. 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	Push-button

An eight-sided star, called an octagram, with Schläfli symbol {8/3} is contained with a regular octagon. A push-button (often simply " button " or " pushbutton " is a simple Switch mechanism for controlling some aspect of a Machine Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one	The vertex figure of the uniform polyhedron, great dirhombicosidodecahedron is contained within an irregular 8-sided star polygon, with four edges going through its center. In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i In Geometry, the great dirhombicosidodecahedron is a nonconvex Uniform polyhedron, indexed last as U75	An octagonal prism contains two octagons. In Geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps
The truncated square tiling has 2 octagons around every vertex. In Geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane	The truncated cuboctahedron has 6 octagons	An octagonal antiprism contains two octagons. The truncated cuboctahedron is an Archimedean solid. It has 12 regular square faces 8 regular Hexagonal faces 6 regular Octagonal faces 48 vertices 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

See also

• octagonal number

External links

• How to find the area of an octagon
• Definition and properties of an octagon With interactive animation
• Eric W. Weisstein, Octagon at MathWorld. An octagonal number is a Figurate number that represents an Octagon. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W