# Dictionary Definition

octagon n : an eight-sided polygon

# User Contributed Dictionary

## English

### Etymology

First attested in 1656, from octagonos, from οκτάγωνος (oktágōnos), eight-angled, from ὀκτώ (okto), eight + γωνία (gonia), angle.### Noun

- A polygon with eight sides and eight angles.

#### Translations

polygon with eight sides and eight angles

- Bulgarian: осмоъгълник
- Czech: osmiúhelník
- Dutch: achthoek
- Finnish: kahdeksankulmio
- French: octogone
- German: Achteck
- Hebrew:
- Icelandic: átthyrningur
- Italian: ottagono
- Japanese: 八角形 (はっかくけい, hakkakukei)
- Russian: восьмиугольник
- Slovak: osemuholník
- Spanish: octágono
- Swedish: oktagon

### Related terms

# Extensive Definition

In geometry, an octagon is a
polygon that has
eight
sides. A regular octagon is represented by the Schläfli
symbol .

## 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 = 2 \cot \frac a^2 = 2(1+\sqrt)a^2 \simeq 4.828427 a^2.

- A = 4 \sin \frac R^2 = 2\sqrtR^2 \simeq 2.828427 R^2.

- A = 8 \tan \frac r^2 = 8(\sqrt-1)r^2 \simeq 3.3137085 r^2.

The area may also be found this way:

- A=S^2-B^2.

Given the span S the length of a side B is

- B = S/(1+\sqrt)./ ,S1 + S2 + S3 + S4 + S5 + S6 +S7 + S8