# Isosceles triangle

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Isosceles triangle
Rank2
Notation
Bowers style acronymIsot
Coxeter diagramox&#y
Elements
Edges1+2
Vertices1+2
Vertex figureDyad
Measures (edge lengths b  [base], l  [legs])
Circumradius${\displaystyle {\frac {l^{2}}{2{\sqrt {l^{2}-{\frac {b^{2}}{4}}}}}}}$
Area${\displaystyle {\frac {b}{2}}{\sqrt {l^{2}-{\frac {b^{2}}{4}}}}}$
Anglesbase–leg: ${\displaystyle \arccos \left({\frac {b}{2l}}\right)}$
leg–leg: ${\displaystyle 2\arcsin \left({\frac {b}{2l}}\right)}$
Height${\displaystyle {\sqrt {l^{2}-{\frac {b^{2}}{4}}}}}$
Central density1
Related polytopes
ArmyIsot
RegimentIsot
DualIsosceles triangle
ConjugateNone
Abstract & topological properties
Flag count6
Euler characteristic0
OrientableYes
Properties
SymmetryA1×I, order 2
ConvexYes
NatureTame

The isosceles triangle, or isot, is a type of triangle with two of the three side lengths equal, as are two of the three angles. It can be considered as a pyramid where the base is a dyad.

Its unequal side is often called its base, in analogy to the pyramid construction. Its equal sides are called its legs.

## Vertex coordinates

Coordinates for an isosceles triangle with base length b and leg length ℓ are given by:

• b/2, 0),
• (0, 2b2/4).

## In vertex figures

All regular polygonal prisms have an isosceles triangle for a vertex figure. Polyhedra with isosceles vertex figures are sometimes known as truncated regular polyhedra, as they all derive from truncations of other polyhedra.[1]

Isosceles triangles in vertex figures
Name Picture Edge lengths
Truncated tetrahedron 1, 3, 3
Truncated cube 1, 2+2, 2+2
Truncated octahedron 2, 3, 3
Quasitruncated hexahedron 1, 2–2, 2–2
Truncated dodecahedron 1, (5+5)/2, (5+5)/2
Truncated icosahedron (5+1)/2, 3, 3
Truncated great dodecahedron (5–1)/2, (5+5)/2, (5+5)/2
Truncated great icosahedron (5–1)/2, 3, 3
Quasitruncated small stellated dodecahedron (5+1)/2, (5–5)/2, (5–5)/2
Quasitruncated great stellated dodecahedron 1, (5–5)/2, (5–5)/2
Truncated square tiling 2, 2+2, 2+2
Truncated hexagonal tiling 1, (2+6)/2, (2+6)/2
Quasitruncated square tiling 2, 2–2, 2–2
Quasitruncated hexagonal tiling 1, (62)/2, (62)/2

## References

1. McCooey, David I. "Self-Intersecting Truncated Regular Polyhedra".