# Scalene triangle

Scalene triangle
Rank2
SpaceSpherical
Bowers style acronymScalene
Info
Coxeter diagramooo&#(a,b,c)
SymmetryI×I, order 1
ArmyScalene
Elements
Edges1+1+1
Vertices1+1+1
Central density1
Euler characteristic0
Related polytopes
DualScalene triangle
ConjugateScalene triangle
Properties
ConvexYes
OrientableYes
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. The term can more widely be used for triangles without any symmetries (other than the identity).

## 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]

${\displaystyle A=\frac14\sqrt{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}.}$

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

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

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:

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

## In vertex figures

Scalene triangles occur as vertex figures of 7 omnitruncated polyhedra.

Scalene triangles in vertex figures
Name Picture Edge lengths
Great 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