# Dictionary Definition

Pentagon

### Noun

1 a government building with five sides that
serves as the headquarters of the United States Department of
Defense

2 the United States military establishment

3 a five-sided polygon

# User Contributed Dictionary

see Pentagon

## English

### Pronunciation

- (US)
- /ˈpɛntəgɑn/
- /"pEnt@gAn/

### Etymology

From French pentagone or late Latin pentagonum, from Greek πεντάγωνον, noun use of the neuter of the adjective πεντάγωνος ‘five-angled’, from πέντε ‘five’ + -γωνος ‘angled’.### Noun

- A polygon with five sides and five angles.

#### Translations

A polygon with five sides and five angles

- Danish: pentagon, femkant
- Dutch: vijfhoek
- Finnish: viisikulmio, pentagoni
- French: pentagone
- German: Fünfeck, Pentagon
- Hebrew: מחומש
- Icelandic: fimmhyrningur
- Irish: peinteagán
- Italian: pentagono
- Japanese: 五角形
- Latvian: piecstūris v.
- Polish: pięciokąt
- Portuguese: pentágono
- Russian: пяти́угольник
- Spanish: pentágono
- Swedish: pentagon

### Derived terms

### See also

# Extensive Definition

In geometry, a pentagon is any
five-sided polygon. A
pentagon may be simple or self-intersecting. The internal
angles in a simple pentagon total 540°.

## Regular pentagons

The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to 108°). Its Schläfli symbol is .The area of a regular convex pentagon with side
length t is given by A = \frac = \frac\ \approx 1.720477401
t^2.

A pentagram is a regular
star
pentagon. Its Schläfli
symbol is . Its sides form the diagonals of a regular convex
pentagon - in this arrangement the sides of
the two pentagons are in the golden
ratio.

### Construction

A regular pentagon is constructible
using a compass
and straightedge, either by inscribing one in a given circle or
constructing one on a given edge. This process was described by
Euclid in
his Elements
circa 300 BC.

One method to construct a regular pentagon in a
given circle is as follows:

An alternative method is this:

- Draw a circle in which to inscribe the pentagon and mark the center point O. (This is the green circle in the diagram to the right).
- Choose a point A on the circle that will serve as one vertex of the pentagon. Draw a line through O and A.
- Construct a line perpendicular to the line OA passing through O. Mark its intersection with one side of the circle as the point B.
- Construct the point C as the midpoint of O and B.
- Draw a circle centered at C through the point A. Mark its intersection with the line OB (inside the original circle) as the point D.
- Draw a circle centered at A through the point D. Mark its intersections with the original (green) circle as the points E and F.
- Draw a circle centered at E through the point A. Mark its other intersection with the original circle as the point G.
- Draw a circle centered at F through the point A. Mark its other intersection with the original circle as the point H.
- Construct the regular pentagon AEGHF.

After forming a regular convex pentagon, if you
join the non-adjacent corners (drawing the diagonals of the
pentagon), you obtain a pentagram, with a smaller
regular pentagon in the center. Or if you extend the sides until
the non-adjacent ones meet, you obtain a larger pentagram.

A simple method of creating a regular pentagon
from just a strip of paper is by tying an overhand
knot into the strip and carefully flattening the knot by
pulling the ends of the paper strip. Folding one of the ends back
over the pentagon will reveal a pentagram when backlit.

## Pentagons in nature

okra
(also called "ladies fingers")

## See also

- Trigonometric constants for a pentagon
- Pentagram
- The Pentagon
- Pentastar
- Dodecahedron, a polyhedron whose regular form is composed of 12 pentagonal faces

## External links

- How to construct a regular pentagon using only compass and straightedge
- How to fold a regular pentagon using only a strip of paper
- Definition and properties of the pentagon, with interactive animation
- Nine constructions for the regular pentagon by Robin Hu
- Renaissance artists' approximate constructions of regular pentagons at Convergence

pentagon in Asturian: Pentágonu

pentagon in Azerbaijani: Düzgün beşbucaqlı

pentagon in Catalan: Pentàgon

pentagon in Czech: Pětiúhelník

pentagon in Danish: Femkant

pentagon in German: Fünfeck

pentagon in Spanish: Pentágono

pentagon in Esperanto: Kvinlatero

pentagon in French: Pentagone (figure)

pentagon in Galician: Pentágono
(xeometría)

pentagon in Korean: 오각형

pentagon in Indonesian: Segi lima

pentagon in Icelandic: Fimmhyrningur

pentagon in Italian: Pentagono (geometria)

pentagon in Hebrew: מחומש

pentagon in Haitian: Pentagòn

pentagon in Hungarian: Ötszög

pentagon in Dutch: Vijfhoek

pentagon in Japanese: 五角形

pentagon in Norwegian: Pentagon

pentagon in Narom: Chîn-carres

pentagon in Central Khmer: បញ្ចកោណ

pentagon in Polish: Pięciokąt

pentagon in Portuguese: Pentágono

pentagon in Romanian: Pentagon (geometrie)

pentagon in Russian: Пятиугольник

pentagon in Simple English: Pentagon

pentagon in Slovak: Päťuholník

pentagon in Slovenian: Petkotnik

pentagon in Serbian: Петоугао

pentagon in Finnish: Viisikulmio

pentagon in Swedish: Pentagon

pentagon in Telugu: పంచభుజి

pentagon in Thai: รูปห้าเหลี่ยม

pentagon in Turkish: Beşgen

pentagon in Yiddish: פינפעק

pentagon in Chinese: 五边形