Isosceles trapezoid

The isosceles trapezoid is a trapezoid with a single symmetry axis. Equivalently, it is a trapezoid with the same leg lengths and base angles. It has 1 top edge, 2 side edges of the same length, and 1 base edge. It has 2 pairs of identical vertices, as the angles at either end of the bases are the same.

Isosceles trapezoid
Rank	2
Space	Spherical
Notation
Bowers style acronym	Traz
Coxeter diagram	xy&#z
Elements
Edges	1+1+2
Vertices	2+2
Vertex figure	Dyad
Measures (edge lengths $b_1$ (small base), $b_2$ (large base), ℓ (lacing))
Area	$\frac{b_1+b_2}{2}\sqrt{l^2-\frac{b_2-b_1}{2}^2}$
Height	$\sqrt{l^2-\frac{b_2-b_1}{2}^2}$
Central density	1
Related polytopes
Army	Traz
Dual	Kite
Conjugate	Isosceles trapezoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A1×I, order 2
Convex	Yes
Nature	Tame

If the legs intersect, the figure may more precisely be called a crossed isosceles trapezoid.

In vertex figures

Every polygonal antiprism has an isosceles trapezoid as its vertex figure. The vertex figures for the crossed antiprisms are themselves crossed, and thus crossed isosceles trapezoids. Seven other uniform polyhedra have trapezoidal vertex figures.

Isosceles trapezoids in vertex figures

Name	Picture	Edge lengths
Triangular antiprism		1, 1, 1, 1
Square antiprism		1, 1, 1, √2
Pentagonal antiprism		1, 1, 1, (√5+1)/2
Pentagrammic antiprism		1, 1, 1, (√5–1)/2
Hexagonal antiprism		1, 1, 1, √3
...
Small rhombicuboctahedron		1, √2, √2, √2
Great cubicuboctahedron		1, √2-√2, √2, √2-√2
Small rhombicosidodecahedron		1, √2, (1+√5)/2, √2
Great dodecicosidodecahedron		1, √(5-√5)/2, (√5-1)/2, √(5-√5)/2
Rhombidodecadodecahedron		(√5-1)/2, √2, (1+√5)/2, √2
Small icosicosidodecahedron		1, √3, (√5-1)/2, √3
Great ditrigonal dodecicosidodecahedron		1, √(5–√5)/2, (1+√5)/2, √(5–√5)/2