Hexacontagone

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Un hexacontagone est un polygone à 60 sommets, donc 60 côtés et 1 710 diagonales.

La somme des angles internes d'un hexacontagone non croisé vaut 10 440 degrés.

L'hexacontagone régulier est constructible.

Hexacontagones réguliers

Un hexacontagone régulier est un hexacontagone dont les côtés ont même longueur et dont les angles internes ont même mesure. Il y en a huit : sept étoilés (notés {60/k} pour k impair de 7 à 29 sauf les multiples de 3 ou 5) et un convexe (noté {60}). C'est de ce dernier qu'il s'agit lorsqu'on parle de « l'hexacontagone régulier ».

Les huit hexacontagones réguliers avec, pour chacun, le symbole de Schläfli et l'angle interne.
{60}, 174°
{60/7}, 138°
{60/11}, 114°
{60/13}, 102°
{60/17}, 78°
{60/19}, 66°
{60/23}, 42°
{60/29}, 6°

Caractéristiques de l'hexacontagone régulier

Chacun des 60 angles au centre mesure ${\frac {360^{\circ }}{60}}=6^{\circ }$ et chaque angle interne mesure ${\frac {10\,440^{\circ }}{60}}=174^{\circ }$ .

Si $a$ est la longueur d'une arête :

le périmètre vaut $P=60\,a$ ;
l'aire vaut $A=15\,a^{2}\cot \left({\frac {\pi }{60}}\right)$ ;
l'apothème vaut $H={\frac {2\,A}{P}}={\frac {a}{2}}\cot \left({\frac {\pi }{60}}\right)$ ;
le rayon vaut $R={\frac {H}{\cos \left({\frac {\pi }{60}}\right)}}={\frac {a}{2\sin \left({\frac {\pi }{60}}\right)}}$ .

Constructibilité

L'hexacontagone régulier est constructible à la règle et au compas, par exemple par bissection du triacontagone.

On pouvait le prévoir grâce au théorème de Gauss-Wantzel, puisque 60 est le produit de 4 (puissance de 2) par 3 et 5 (nombres premiers de Fermat distincts).

Articles connexes

Expression des lignes trigonométriques pour les premiers multiples de 3° = $π/60$ rad

v · m

Polygones

Triangles

Quadrilatères

Par nombre de côtés

1 à 10 côtés	Hénagone (1) Digone (2) Triangle (3) Quadrilatère (4) Pentagone (5) Hexagone (6) Heptagone (7) Octogone (8) Ennéagone (9) Décagone (10)
11 à 20 côtés	Hendécagone (11) Dodécagone (12) Tridécagone (13) Tétradécagone (14) Pentadécagone (15) Hexadécagone (16) Heptadécagone (17) Octadécagone (18) Ennéadécagone (19) Icosagone (20)
30 côtés et plus	Triacontagone (30) Tétracontagone (40) Pentacontagone (50) Hexacontagone (60) Heptacontagone (70) Octacontagone (80) Ennéacontagone (90) Hectogone (100) Dihectogone (200) Trihectogone (300) Tétrahectogone (400) Pentahectogone (500) Hexahectogone (600) Heptahectogone (700) Octahectogone (800) Ennéahectogone (900) Chiliogone (1 000) Myriagone (10 000) Mégagone (en) (1 000 000) Apeirogone (∞)

Autres classements que par le nombre des côtés

Classement par convexité
- Polygone croisé
- Polygone simple
  - Polygone non convexe
    - Polygone étoilé
  - Polygone convexe
Classement par les angles et les côtés
Classement par rapport à un cercle

Polygones réguliers étoilés

Description

Droites et cercles remarquables

Relations entre polygones

Construction

Dissection

Portail de la géométrie

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