# Scalene triangle

Scalene triangle Rank2
SpaceSpherical
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
ConjugateScalene triangle
Abstract properties
Euler characteristic0
Topological properties
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:

$A=\frac14\sqrt{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}.$ The law of sines allows one to then derive the following expression for the circumradius:

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

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

Scalene triangles occur as vertex figures of 7 omnitruncated polyhedra.