The Square Pyramid has 5 vertices, 8 edges and 5 faces: 4 triangles, 1 square.

To construct the Square Pyramid, you need:
5
8 (G0, G1 or G2 size)

The Pentagonal Pyramid has 6 vertices, 10 edges and 6 faces: 5 triangles, 1 pentagon.

To construct the Pentagonal Pyramid, you need:
6
10 (B0, B1 or B2 size)

The Triangular Cupola has 9 vertices, 15 edges and 8 faces: 4 triangles, 3 squares, 1 hexagon.

To construct the Triangular Cupola, you need:
9
15 (G0, G1 or G2 size)

The Square Cupola has 12 vertices, 20 edges and 10 faces: 4 triangles, 5 squares, 1 octagon.

To construct the Square Cupola as close approximation, you need:
12
8 (B2)
12 (G1)

The Pentagonal Cupola has 15 vertices, 25 edges and 12 faces: 5 triangles, 5 squares, 1 pentagon, 1 decagon.

To construct the Pentagonal Cupola, you need:
15
25 (B0, B1 or B2 size)

The Pentagonal Rotunda has 20 vertices, 35 edges and 17 faces: 10 triangles, 6 pentagons, 1 decagon.

To construct the Pentagonal Rotunda, you need:
20
35 (B0, B1 or B2 size)

The Elongated Triangular Pyramid has 7 vertices, 12 edges and 7 faces: 4 triangles, 3 squares.

To construct the Elongated Triangular Pyramid as close approximation, you need:
7
3 (Y2)
9 (G1)

The Elongated Square Pyramid has 9 vertices, 16 edges and 9 faces: 4 triangles, 5 squares.

To construct the Elongated Square Pyramid as close approximation, you need:
9
4 (B1)
12 (G1)

The Elongated Pentagonal Pyramid has 11 vertices, 20 edges and 11 faces: 5 triangles, 5 squares, 1 pentagon.

To construct the Elongated Pentagonal Pyramid as close approximation, you need:
11
15 (B1)
5 (R1)

The Gyroelongated Pentagonal Pyramid (Diminished Icosahedron) has 11 vertices, 25 edges and 16 faces: 15 triangles, 1 pentagon.

To construct the Gyroelongated Pentagonal Pyramid, you need:
11
25 (B0, B1 or B2 size)

The Pentagonal Bipyramid has 7 vertices, 15 edges and 10 faces: 10 triangles.

To construct the Pentagonal Bipyramid, you need:
7
10 (B0, B1 or B2 size)
5 (R0, R1 or R2 size)

The Elongated Square Bipyramid has 10 vertices, 20 edges and 12 faces: 8 triangles, 4 squares.

To construct the Elongated Square Bipyramid, you need:
10
4 (B0, B1 or B2 size)
16 (G0, G1 or G2 size)

The Elongated Pentagonal Bipyramid has 12 vertices, 25 edges and 15 faces: 10 triangles, 5 squares.

To construct the Elongated Pentagonal Bipyramid, you need:
12
15 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Elongated Triangular Cupola has 15 vertices, 27 edges and 14 faces: 4 triangles, 9 squares, 1 hexagon.

To construct the Elongated Triangular Cupola, you need:
15
6 (Y0, Y1 or Y2 size)
21 (G0, G1 or G2 size)

The Elongated Square Cupola has 20 vertices, 36 edges and x faces: 4 triangles, 13 squares, 1 octagon.

To construct the Elongated square cupola, you need:
20
18 (B0, B1 or B2 size)
16 (G0, G1 or G2 size)

The Elongated Pentagonal Cupola has 25 vertices, 45 edges and 22 faces: 5 triangles, 15 squares, 1 pentagon, 1 decagon.

To construct the Elongated Pentagonal Cupola, you need:
25
35 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Elongated Pentagonal Rotunda has 30 vertices, 55 edges and 27 faces: 10 triangles, 10 squares, 6 pentagons, 1 decagon.

To construct the Elongated Pentagonal Rotunda, you need:
30
45 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Square Orthobicupola has 16 vertices, 32 edges and 18 faces: 8 triangles, 10 squares.

To construct the Square Orthobicupola, you need:
16
12 (B0, B1 or B2 size)
20 (G0, G1 or G2 size)

The Pentagonal Orthobicupola has 20 vertices, 40 edges and 22 faces: 10 triangles, 10 squares, 2 pentagons.

To construct the Pentagonal Orthobicupola, you need:
20
30 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Pentagonal Orthocupolarotunda has 25 vertices, 50 edges and 30 faces: 15 triangles, 5 squares, 7 pentagons.

To construct the Pentagonal Orthocupolarotunda, you need:
25
50 (B0, B1 or B2 size)

The Pentagonal Gyrocupolarotunda has 25 vertices, 50 edges and 30 faces: 15 triangles, 5 squares, 10 pentagons.

To construct the Pentagonal Gyrocupolarotunda, you need:
25
40 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Pentagonal Orthobirotunda has 30 vertices, 60 edges and 32 faces: 20 triangles, 12 pentagons.

To construct the Pentagonal Orthobirotunda, you need:
x
40 (B0, B1 or B2 size)
10 (Y0, Y1 or Y2 size)
10 (R0, R1 or R2 size)

The Elongated Pentagonal Orthobicupola has 30 vertices, 60 edges and 32 faces: 10 triangles, 20 squares, 2 pentagons.

To construct the Elongated Pentagonal Orthobicupola, you need:
30
40 (B0, B1 or B2 size)
20 (R0, R1 or R2 size)

The Elongated Pentagonal Gyrobicupola has 30 vertices, 60 edges and 32 faces: 10 triangles, 20 squares, 2 pentagons.

To construct the Elongated Pentagonal Gyrobicupola, you need:
30
50 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Elongated Pentagonal Orthocupolarotunda has 35 vertices, 70 edges and 37 faces: 15 triangles, 15 squares, 7 pentagons.

To construct the Elongated Pentagonal Orthocupolarotunda, you need:
35
60 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Elongated Pentagonal Gyrocupolarotunda has 35 vertices, 70 edges and 37 faces: 15 triangles, 15 squares, 7 pentagons.

To construct the Elongated Pentagonal Gyrocupolarotunda, you need:
35
50 (B0, B1 or B2 size)
20 (R0, R1 or R2 size)

The Elongated Pentagonal Orthobirotunda has 40 vertices, 80 edges and 42 faces: 20 triangles, 10 squares, 12 pentagons.

To construct the Elongated Pentagonal Orthobirotunda, you need:
40
50 (B0, B1 or B2 size)
10 (Y0, Y1 or Y2 size)
20 (R0, R1 or R2 size)

The Elongated Pentagonal Gyrobirotunda has 40 vertices, 80 edges and 42 faces: 20 triangles, 10 squares, 12 pentagons.

To construct the Elongated Pentagonal Gyrobirotunda, you need:
40
70 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Augmented Dodecahedron has 21 vertices, 35 edges and 16 faces: 5 triangles, 11 pentagons.

To construct the Augmented Dodecahedron, you need:
21
30 (B0, B1 or B2 size)
5 (R0, R1 or R2 size)

The Parabiaugmented Dodecahedron has 22 vertices, 40 edges and 20 faces: 10 triangles, 10 pentagons.

To construct the Parabiaugmented Dodecahedron, you need:
22
30 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Metabiaugmented Dodecahedron has 22 vertices, 40 edges and 20 faces: 10 triangles, 10 pentagons.

To construct the Metabiaugmented Dodecahedron, you need:
22
30 (B0, B1 or B2 size)
10 (R0, R1 or R2 size)

The Triaugmented Dodecahedron has 23 vertices, 45 edges and 24 faces: 15 triangles, 9 pentagons.

To construct the Triaugmented Dodecahedron, you need:
23
30 (B0, B1 or B2 size)
15 (R0, R1 or R2 size)

The Metabidiminished Icosahedron has 10 vertices, 20 edges and 12 faces: 10 triangles, 2 pentagons.

To construct the Metabidiminished Icosahedron, you need:
10
20 (B0, B1 or B2 size)

The Tridiminished Icosahedron has 9 vertices, 15 edges and 8 faces: 5 triangles, 3 pentagons.

To construct the Tridiminished Icosahedron, you need:
9
15 (B0, B1 or B2 size)

The Augmented Tridiminished Icosahedron has 10 vertices, 18 edges and 10 faces: 7 triangles, 3 pentagons.

To construct the Augmented Tridiminished Icosahedron, you need:
10
15 (B0, B1 or B2 size)
3 (Y0, Y1 or Y2 size)

The Augmented Truncated Tetrahedron has 15 vertices, 27 edges and 14 faces: 8 triangles, 3 squares, 3 hexagons.

To construct the Augmented Truncated Tetrahedron, you need:
15
27 (G0, G1 or G2 size)

The Augmented Truncated Cube has 28 vertices, 48 edges and 22 faces: 12 triangles, 5 squares, 5 octagons.

To construct the Augmented Truncated Cube, you need:
28
16 (B0, B1 or B2 size)
32 (G0, G1 or G2 size)

The Biaugmented Truncated Cube has 32 vertices, 60 edges and 30 faces: 16 triangles, 10 squares, 4 octagons.

To construct the Biaugmented Truncated Cube, you need:
32
20 (B0, B1 or B2 size)
40 (G0, G1 or G2 size)

The Diminished Rhombicosidodecahedron has 55 vertices, 105 edges and 52 faces: 15 triangles, 25 squares, 11 pentagons, 1 decagon.

To construct the Diminished Rhombicosidodecahedron, you need:
55x
105 (B0, B1 or B2 size)

The Parabidiminished Rhombicosidodecahedron has 50 vertices, 90 edges and 42 faces: 10 triangles, 20 squares, 10 pentagons, 2 decagons.

To construct the Parabidiminished Rhombicosidodecahedron, you need:
50
90 (B0, B1 or B2 size)

The Metabidiminished Rhombicosidodecahedron has 50 vertices, 90 edges and 42 faces: 10 triangles, 20 squares, 10 pentagons, 2 decagons.

To construct the Metabidiminished Rhombicosidodecahedron, you need:
50
90 (B0, B1 or B2 size)

The Tridiminished Rhombicosidodecahedron has 45 vertices, 75 edges and 32 faces: 5 triangles, 15 squares, 9 pentagons, 3 decagons.

To construct the Tridiminished Rhombicosidodecahedron, you need:
45
75 (B0, B1 or B2 size)

The Sphenocorona has 10 vertices, 22 edges and 14 faces: 12 triangles, 2 squares.

To construct the Sphenocorona, you need:
10
18 (B0, B1 or B2 size)
6 (Y0, Y1 or Y2 size)

The Bilunabirotunda has 14 vertices, 26 edges and 14 faces: 8 triangles, 2 squares, 4 pentagons. See this YouTube video for the construction and animated views of the Bilunabirotunda.

To construct the Bilunabirotunda, you need:
14
26 (B0, B1 or B2 size)

The Triangular Hebesphenorotunda has 18 vertices, 36 edges and 20 faces: 13 triangles, 3 squares, 3 pentagons, 1 hexagon.

To construct the Triangular Hebesphenorotunda, you need:
18
36 (B0, B1 or B2 size)

The "6 Bilunabirotundae around a cube" has 56 vertices, 120 edges and 54 faces: 24 triangles, 6 squares, 24 pentagons. See this YouTube video for the construction and animated views of the 6 Bilunabirotundae model.

To construct the "6 Bilunabirotundae around a cube", you need:
56
120 (B0, B1 or B2 size)