Octagon

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For other uses, see Octagon (disambiguation).

A regular octagon.

In many parts of the world, stop signs are regular octagons

In geometry, an octagon is a polygon that has eight sides.

In some parts of the world, stop signs are in the shape of a regular octagon.

1 Regular octagons
- 1.1 Construction
2 See also
3 External links

[edit] Regular octagons

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

$A = 2a^2 \cot \frac{\pi}{8} = 2(1+\sqrt{2})a^2 \simeq 4.82843 a^2.$

In terms of $R$ , (circumradius) the area is $8R^2/(1+\sqrt{2})$ ^{[citation needed]}

In terms of $r$ , (inradius) the area is $8r^2/(11+9\sqrt{2})$
The area may also be found this way:

A = S 2 - B 2

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. The edges of this square are each the length of the base.

[edit] Construction

An octagon is constructible with compass and straightedge. To do so, follow steps 1 through 18 of the animation, noting that the compass radius is not altered during steps 7 though 10.

[edit] See also

octagonal number

[edit] External links

How to find the area of an octagon
Definition and properties of an octagon With interactive animation

Polygons
Triangle • Quadrilateral • Pentagon •Hexagon • Heptagon • Octagon • Enneagon (Nonagon) • Decagon • Hendecagon • Dodecagon • Triskaidecagon • Pentadecagon • Hexadecagon • Heptadecagon • Enneadecagon • Icosagon • Chiliagon • Myriagon