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   Regular Polyhedra: Formulas   

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also see Defining Geometric Figures

  Regular Polyhedron

    A solid, three-dimensional 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 three-dimensional 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)a2
V = (sqrt[2]/12)a3
delta = arccos(1/3) = 70o 32'

h = height or altitude
h = (sqrt[6]/3)a




  Cube

 
 

   
A three-dimensional 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 a2
V = a3
delta = arccos(0) = 90o

See Ask Dr. Math: Surface Area and Volume: Cubes and Prisms



  Octahedron

 
 

   
A three-dimensional 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)a2
V = (sqrt[2]/3)a3
delta = arccos(-1/3) = 109o 28'



  Dodecahedron

 
 

   
A three-dimensional 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])a2
V = ([15+7 sqrt(5)]/4)a3
delta = arccos(-sqrt[5]/5) = 116o 34'



  Icosahedron

 
 

   
A three-dimensional figure with
20 equilateral triangle faces,
12 vertices, and 30 edges.

     
v = 12, e = 30, f = 20
a = (sqrt[50-10 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)a2
V = (5[3+sqrt(5)]/12)a3
delta = arccos(-sqrt[5]/3) = 138o 11'

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