Scalene triangle

Scalene triangle
Rank	2
Space	Spherical
Notation
Bowers style acronym	Scalene
Coxeter diagram	ooo&#(a,b,c)
Elements
Edges	1+1+1
Vertices	1+1+1
Vertex figure	Dyad
Measures (edge lengths a, b, c)
Central density	1
Related polytopes
Army	Scalene
Dual	Scalene triangle
Conjugate	Scalene triangle
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	I×I, order 1
Convex	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.

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.

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
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

External links

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