Simplest Canonical Polyhedra of Each Symmetry Type

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

Simplest Canonical Polyhedron with C10v Symmetry
(Decagonal Pyramid)
Vertices:  11  (10[3] + 1[10])
Faces:11  (10 isosceles triangles + 1 regular decagon)
Edges:20  (10 short + 10 long)
Symmetry:  10-fold Pyramidal  (C10v)
Dual Solid:  Decagonal Pyramid (itself)
(values below based on edge-scribed radius = 1)
Short Edge (10):  2*sqrt(10*(sqrt(10*(5−sqrt(5)))−5))/5    ≈0.64163680840966230871
Long Edge (10):  2*sqrt(5*(15+5*sqrt(5)    
    +4*sqrt(5*(5+2*sqrt(5)))))/5    
≈6.554874723252759115003
Edge-scribed Radius:  1
Volume:40*(6+2*sqrt(5)−sqrt(2*(25+11*sqrt(5))))/3    ≈6.8339042062461106562