Scalene triangle

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Scalene triangle
Scalene triangle.svg
Bowers style acronymScalene
Coxeter diagramooo&#(a,b,c)
Vertex figureDyad
Measures (edge lengths a, b, c)
Central density1
Related polytopes
DualScalene triangle
ConjugateScalene triangle
Abstract properties
Euler characteristic0
Topological properties
SymmetryI×I, order 1

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[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]