Regular octagon | |
---|---|
A regular octagon | |
Edges and vertices | 8 |
Schläfli symbols | {8} t{4} |
Coxeter–Dynkin diagrams | |
Symmetry group | Dihedral (D8) |
Area (with a=edge length) |
|
Internal angle (degrees) |
135° |
Template:Otheruses
In geometry, an octagon is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.
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 at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080°. The area of a regular octagon of side length is given by
In terms of , (circumradius) the area is
In terms of , (inradius) the area is
Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.
The area can also be derived as folllows:
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, is then as above:
Uses of octagons[]
Derived figures[]
Petrie polygons[]
The octagon is the Petrie polygon for these 12 higher-dimensional uniform polytopes, shown in these skew orthogonal projections of in A7, B4, and D5 Coxeter planes.
A7 | 7-simplex |
Rectified 7-simplex |
Birectified 7-simplex |
Trirectified 7-simplex |
---|---|---|---|---|
B4 | 16-cell |
Rectified 16-cell |
Rectified tesseract |
Tesseract |
D5 | Trirectified 5-demicube |
Birectified 5-demicube |
Rectified 5-demicube |
5-demicube |
See also[]
- Octagram
- Octagonal number
- Octagon house
- Bumper pool
- Leonardo da Vinci
External links[]
- How to find the area of an octagon
- Definition and properties of an octagon With interactive animation
- Weisstein, Eric W., "Octagon" from MathWorld.
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