Symmetric Canonical Self-Dual Pentahedra

This is the only one

(box: x-ray)  (F: faces)  (slider: perspective)  (image: L=rotate R=zoom)  (M: metrics)

Self-Dual Pentahedron (canonical)
Vertices:  5  (4[3] + 1[4])
Faces:5  (4 isosceles triangles + 1 square)
Edges:8  (4 short + 4 long)
Symmetry:  4-fold Pyramidal  (C4v)
(values below based on edge-scribed radius = 1)
Short Edge (4):  2*sqrt(2*(sqrt(2)−1))    ≈1.8203594422489093652
Long Edge (4):  2*sqrt(1+sqrt(2))    ≈3.1075479480600746147
Edge-scribed Radius:  1
Volume:16*(2−sqrt(2))/3    ≈3.1241943340101597397