A regular octagon
|Edges and vertices||8|
|Symmetry group||Dihedral (D8)|
(with a=edge length)
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
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
- 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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