# Scalene triangle

Scalene triangle
Rank2
Notation
Bowers style acronymScalene
Coxeter diagramooo&#(a,b,c)
Elements
Edges1+1+1
Vertices1+1+1
Measures (edge lengths a, b, c)
Central density1
Related polytopes
ArmyScalene
DualScalene triangle
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI×I, order 1
ConvexYes
NatureTame

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.

## Measures

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

${\displaystyle A={\frac {1}{4}}{\sqrt {2a^{2}b^{2}+2b^{2}c^{2}+2c^{2}a^{2}-a^{4}-b^{4}-c^{4}}}.}$

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

${\displaystyle R={\frac {abc}{\sqrt {2a^{2}b^{2}+2b^{2}c^{2}+2c^{2}a^{2}-a^{4}-b^{4}-c^{4}}}}.}$

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:

${\displaystyle \alpha ={\text{asin}}\left({\frac {\sqrt {2a^{2}b^{2}+2b^{2}c^{2}+2c^{2}a^{2}-a^{4}-b^{4}-c^{4}}}{bc}}\right),}$
${\displaystyle \beta ={\text{asin}}\left({\frac {\sqrt {2a^{2}b^{2}+2b^{2}c^{2}+2c^{2}a^{2}-a^{4}-b^{4}-c^{4}}}{ca}}\right),}$
${\displaystyle \gamma ={\text{asin}}\left({\frac {\sqrt {2a^{2}b^{2}+2b^{2}c^{2}+2c^{2}a^{2}-a^{4}-b^{4}-c^{4}}}{ab}}\right).}$

## In vertex figures

Scalene triangles occur as vertex figures of 7 omnitruncated polyhedra.

Scalene triangles in vertex figures
Name Picture Edge lengths
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