# Scalene triangle

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Scalene triangle | |
---|---|

Rank | 2 |

Space | Spherical |

Bowers style acronym | Scalene |

Info | |

Coxeter diagram | ooo&#(a,b,c) |

Symmetry | I×I, order 1 |

Army | Scalene |

Elements | |

Vertex figure | Dyad |

Edges | 1+1+1 |

Vertices | 1+1+1 |

Central density | 1 |

Euler characteristic | 0 |

Related polytopes | |

Dual | Scalene triangle |

Conjugate | Scalene triangle |

Properties | |

Convex | Yes |

Orientable | Yes |

Nature | Tame |

The **scalene triangle**, or **scalene**, is a type of triangle with none of its three side lengths equal, and all of its three angles different. The term can more widely be used for triangles without any symmetries (other than the identity).

## Measures[edit | edit source]

The area of a general triangle with side lengths *a*, *b*, and *c* satisfying the triangle inequality is given by Heron's formula:^{[1]}

The law of sines allows one to then derive the following expression for the circumradius:^{[2]}

The angles *α*, *β*, and *γ* of the triangle, opposite to the sides with lengths *a*, *b*, *c*, respectively, are also given by the law of sines, as:

## In vertex figures[edit | edit source]

Scalene triangles occur as vertex figures of 7 omnitruncated polyhedra.

Name | Picture | Edge lengths |
---|---|---|

Great rhombitetratetrahedron | √2, √3, √3 | |

Great rhombicuboctahedron | √2, √3, √2+√2 | |

Quasitruncated cuboctahedron | √2, √3, √2–√2 | |

Cuboctatruncated cuboctahedron | √3, √2+√2, √2–√2 | |

Great rhombicosidodecahedron | √2, √3, √(5+√5)/2 | |

Quasitruncated dodecadodecahedron | √2, √(5+√5)/2, √(5–√5)/2 | |

Great quasitruncated icosidodecahedron | √2, √3, √(5–√5)/2 | |

Icosidodecatruncated icosidodecahedron | √3, √(5+√5)/2, √(5–√5)/2 |

## References[edit | edit source]

- ↑ Weisstein, Eric W. "Heron's Formula".
*MathWorld*. - ↑ Weisstein, Eric W. "Circumradius".
*MathWorld*.

## External links[edit | edit source]

- Wikipedia Contributors. "Triangle § By lengths of sides".