also see Defining Geometric Figures
Regular Polyhedron
A solid, threedimensional figure each face of
which is a regular polygon with
equal sides and equal angles. Every face has the same
number of vertices, and the
same number of faces meet at every vertex. An inscribed
(inside) sphere touches
the center of every face, and a circumscribed sphere (outside)
touches every vertex.
There are five and only five of these figures, also called the Platonic Solids:
the tetrahedron,
cube,
octahedron,
dodecahedron, and
icosahedron.


Number of vertices: v
Number of edges: e
Number of faces: f
Edge: a
Radius of circumscribed sphere: R
Radius of inscribed sphere: r
Surface area: S
Volume: V
Dihedral angle between faces: delta (in degrees) 


To read about regular polyhedra, visit
The Geometry Center: Regular
Polyhedra
Mathematics Encyclopedia: The
Platonic Solids
Eric Swab: Cube as the Base
To explore polyhedra via an MJA applet, try
Suzanne Alejandre's
Studying Polyhedra
For many more facts and links, see
Alexander Bogomolny's
Regular polyhedra
For a Java applet showing many other polyhedra, see
John N. Huffman's
Crystallographic Polyhedra

Tetrahedron 
(see also Irregular Tetrahedron)


A threedimensional figure with
4 equilateral triangle faces,
4 vertices, and 6 edges.




v = 4, e = 6, f = 4
a = (2 sqrt[6]/3)R
r = (1/3)R
R = (sqrt[6]/4)a
S = sqrt(3)a^{2}
V = (sqrt[2]/12)a^{3}
delta = arccos(1/3) = 70^{o} 32'
h = height or altitude
h = (sqrt[6]/3)a 
Cube 


A threedimensional figure with
6 square faces, 8 vertices,
and 12 edges. 



v = 8, e = 12, f = 6
a = (2 sqrt[3]/3)R
r = (sqrt[3]/3)R
R = (sqrt[3]/2)a
r = (1/2)a
S = 6 a^{2}
V = a^{3}
delta = arccos(0) = 90^{o} 
See Ask Dr. Math: Surface Area and Volume: Cubes and Prisms
Octahedron 


A threedimensional figure with
8 equilateral triangle faces,
6 vertices, and 12 edges. 



v = 6, e = 12, f = 8
a = sqrt(2)R
r = (sqrt[3]/3)R
R = (sqrt[2]/2)a
r = (sqrt[6]/6)a
S = 2 sqrt(3)a^{2}
V = (sqrt[2]/3)a^{3}
delta = arccos(1/3) = 109^{o} 28' 
Dodecahedron 


A threedimensional figure with
12 regular pentagon faces,
20 vertices, and 30 edges. 



v = 20, e = 30, f = 12
a = ([sqrt(5)1]sqrt[3]/3)R
r = (sqrt[75+30 sqrt(5)]/15)R
R = (sqrt[3][1+sqrt[5])/4)a
r = (sqrt[250+110 sqrt(5)]/20)a
S = 3 sqrt(25+10 sqrt[5])a^{2}
V = ([15+7 sqrt(5)]/4)a^{3}
delta = arccos(sqrt[5]/5) = 116^{o} 34' 
Icosahedron 


A threedimensional figure with
20 equilateral triangle faces,
12 vertices, and 30 edges. 



v = 12, e = 30, f = 20
a = (sqrt[5010 sqrt(5)]/5)R
r = (sqrt[75+30 sqrt(5)]/15)R
R = (sqrt[10+2 sqrt(5)]/4)a
r = (sqrt[42+18 sqrt(5)]/12)a
S = 5 sqrt(3)a^{2}
V = (5[3+sqrt(5)]/12)a^{3}
delta = arccos(sqrt[5]/3) = 138^{o} 11' 
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