Scalene triangle

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Scalene triangle
Scalene triangle.svg
Rank2
SpaceSpherical
Bowers style acronymScalene
Info
Coxeter diagramooo&#(a,b,c)
SymmetryI×I, order 1
ArmyScalene
Elements
Vertex figureDyad
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[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.

Scalene triangles in vertex figures
Name Picture Edge lengths
Great rhombitetratetrahedron
Omnitruncated tetrahedron.png
2, 3, 3
Great rhombicuboctahedron
Great rhombicuboctahedron.png
2, 3, 2+2
Quasitruncated cuboctahedron
Great truncated cuboctahedron.png
2, 3, 2–2
Cuboctatruncated cuboctahedron
Cubitruncated cuboctahedron.png
3, 2+2, 2–2
Great rhombicosidodecahedron
Great rhombicosidodecahedron.png
2, 3, (5+5)/2
Quasitruncated dodecadodecahedron
Truncated dodecadodecahedron.png
2, (5+5)/2, (5–5)/2
Great quasitruncated icosidodecahedron
Great truncated icosidodecahedron.png
2, 3, (5–5)/2
Icosidodecatruncated icosidodecahedron
Icositruncated dodecadodecahedron.png
3, (5+5)/2, (5–5)/2

References[edit | edit source]

  1. Weisstein, Eric W. "Heron's Formula". MathWorld.
  2. Weisstein, Eric W. "Circumradius". MathWorld.

External links[edit | edit source]