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# Polyhedra Notes

written by Rene K. Mueller, Copyright (c) 2007, last updated Wed, January 9, 2008

### Square Pyramid

 Square Pyramid
• Johnson Solid: J1
• Vertices: 5
• Faces: 5
• Edges: 8
• V: s3 / 6 * √2
• A: s2 * (1 + √3)

 General Pyramid
The general pyramid:

• q = √((s/2)2 + h2)
• l = √((s/2)2 + q2)
• A = 2 s q + s2
• Awall/roof = 2 s q
• V = s2 h / 3

#### Historic Pyramids

 The Pyramids of Giza (courtesy Bruno Girin/DHD Multimedia Gallery )
Most obvious association with pyramids comes from the egyptian pyramids in Giza and those of the mayans in south america. It is also apparent those pyramids have been built with enormous effort and skills which is now honored by the durability of thousands of years.

The largest pyramid in Giza is the Khufu Pyramid, with following values:

• s = 231m (original) (now: 230.4m)
• h = 146.6m (original) (now: 138.8m)

thereby

• q = 186.632m
• l = 219.478m

Interesting ratio:

• s/h = 1.57571.. ~ π/2 ~ 11/7

Read more at Great Pyramid of Giza .

### Pentagonal Pyramid

 Pentagonal Pyramid
• Johnson Solid: J2
• Vertices: 6
• Faces: 6
• Edges: 10
• V: s3 / 24 * (5 + √5)
• A: s2 / 2 * √(5/2 * (10 + √10 + √(75+30*√5))

The roof isn't very steep, unless those roof struts would be made longer - it would be a very simplistic habitat.

### Triangular Cupola

 Triangular Cupola
• Johnson Solid: J3
• Vertices: 9
• Faces: 8
• Edges: 15
• V: s3 * (5/(5*√2))
• A: s2 * (3+5/2*√3)

### Square Cupola

 Square Cupola
• Johnson Solid: J4
• Vertices: 12
• Faces: 10
• Edges: 20

Also cap of a Rhombicuboctahedron.

### Pentagonal Cupola

 Pentagonal Cupola
• Johnson Solid: J5
• Vertices: 15
• Faces: 12
• Edges: 25
• V: s3 / 6 * (5+4*√5)
• A: s2 / 4 * (20+√(10*(80+31*√5+√(2175+950*√5))))

Also cap of a Rhombicosidodecahedron.

### Pentagonal Rotunda

 Pentagonal Rotunda
• Johnson Solid: J6
• Vertices: 20
• Faces: 17
• Edges: 35
• V: s3 / 12 * (45+17*√5)
• A: s2 * 5/2 * (√3+√(26+(58/√5)))
• h: s * √(1/5*(5+2*√5))
Also 1/2 of a Icosidodecahedron.

### Elongated Triangular Pyramid

 Elongated Triangular Pyramid
• Johnson Solid: J7
• Vertices: 7
• Faces: 7
• Edges: 12

### Elongated Square Pyramid

 Elongated Square Pyramid
• Johnson Solid: J8
• Vertices: 9
• Faces: 9
• Edges: 16

### Elongated Pentagonal Pyramid

 Elongated Pentagonal Pyramid
• Johnson Solid: J9
• Vertices: 11
• Faces: 11
• Edges: 20

### Gyroelongated Square Pyramid

 Gyroelongated Square Pyramid
• Johnson Solid: J10
• Vertices: 9
• Faces: 13
• Edges: 20

### Gyroelongated Pentagonal Pyramid

 Gyroelongated Pentagonal Pyramid
• Johnson Solid: J11
• Vertices: 11
• Faces: 16
• Edges: 25

### Triangular Dipyramid

 Triangular Dipyramid
• Johnson Solid: J12
• Vertices: 5
• Faces: 6
• Edges: 9

### Pentagonal Dipyramid

 Pentagonal Dipyramid
• Johnson Solid: J13
• Vertices: 7
• Faces: 10
• Edges: 15
• V: s3 / 12 * (5 + √5)
• A: s2 * 5/12 * √3
• router: s / 10 * √(50+10*√5)
• h: s / 10 * √(50-10*√5)

### Elongated Triangular Dipyramid

 Elongated Triangular Dipyramid
• Johnson Solid: J14
• Vertices: 8
• Faces: 9
• Edges: 15

### Elongated Square Dipyramid

 Elongated Square Dipyramid
• Johnson Solid: J15
• Vertices: 10
• Faces: 12
• Edges: 20

### Elongated Pentagonal Dipyramid

 Elongated Pentagonal Dipyramid
• Johnson Solid: J16
• Vertices: 12
• Faces: 15
• Edges: 25

### Gyroelongated Square Dipyramid

 Gyroelongated Square Dipyramid
• Johnson Solid: J17
• Vertices: 10
• Faces: 16
• Edges: 24
• V: s3 * 2(1/4) / 4 * (1 + √2 + 2(1/4) )
• A: s2 * 4 * √3

### Elongated Triangular Cupola

 Elongated Triangular Cupola
• Johnson Solid: J18
• Vertices: 15
• Faces: 14
• Edges: 27

### Elongated Square Cupola

 Elongated Square Cupola
• Johnson Solid: J19
• Vertices: 20
• Faces: 18
• Edges: 36

### Elongated Pentagonal Cupola

 Elongated Pentagonal Cupola
• Johnson Solid: J20
• Vertices: 25
• Faces: 22
• Edges: 45

### Elongated Pentagonal Rotunda

 Elongated Pentagonal Rotunda
• Johnson Solid: J21
• Vertices: 30
• Faces: 27
• Edges: 55
Very suitable for a habitat, the top smaller pentagon as skylight, and eventually the side pentagons further triangulated each.

### Gyroelongated Triangular Cupola

 Gyroelongated Triangular Cupola
• Johnson Solid: J22
• Vertices: 15
• Faces: 20
• Edges: 33

### Gyroelongated Square Cupola

 Gyroelongated Square Cupola
• Johnson Solid: J23
• Vertices: 20
• Faces: 26
• Edges: 44

### Gyroelongated Pentagonal Cupola

 Gyroelongated Pentagonal Cupola
• Johnson Solid: J24
• Vertices: 25
• Faces: 32
• Edges: 55

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