REGULAR CONVEX POLYHEDRONS  - Common Properties - One Family






REGULAR CONVEX POLYHEDRONS

*(p,q)


Vertices (v)


Edges (e)


Faces (f)

Vertex angle (α)

Dihedral angle
(
θ)


Inradius (r)


Circumradius (R)


R/r


Surface area (A)


Volume (V)


P

l

a

t

o

n

i

c

   s

o

l

i

d

s


tetrahedron



(3,3)


4



6


4


60°


70.53°





3





cube



(4,3)


8



12


6


90°


90°











octahedron


(3,4)


6


12


8


60°


109.47°











dodecahedron



(5,3)


20


30


12


108°


116.57°




≈ 1,25841






icosahedron


(3,5)


12


30


20


60°


138.19°



≈ 1,25841






S

p

h

e

r

i

c

a

 l 

   p o l y h e d r o n s



dihedrons



(2,n)


2


2…n



2…n



360°/n           










180°










r










r










1
































P

l

a

t

o

n

i

c



s

p

h

e

r

e

s



tetrahedral


(3,3)


4


6


4


120°


octahedral


(3,4)


6


12


8


90°



hexahedral




(4,3)


8


12


6


120°


dodecahedral




(5,3)


20

30


12


120°


icosahedral


(3,5)


12


30


20


72°


                                                               * the number of edges in a face,  q the number of edges meeting at a vertex  
    


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                          (3)                                 v + f - e = 2           Euler's formula
                 

 
                      (4)                           fp = vq = 2e    

                                                                    

  

                           

                                                                                           
                                                                            
                                                                                                    


                                                                                                     By mirco                                                                                                                                   
                                                                                                     Enjoy!



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