Calculations For Polyhedra

 

Definition Of Regular Polyhedron

A regular polyhedron is 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. See Figure 1.

 

 

 Notation:

The following table presents a summary of the mathematical notation used in the paragraphs which follow.

 

Item

Symbol

 

 

Number of vertices

v

Number of edges

e

Number of faces

f

Length of an edge

a

Diameter of circumscribed sphere

d

Radius of circumscribed sphere

R

Radius of inscribed sphere

r

 

 

The Regular Plane Figures

 

Since the faces of all five of these figures consist of either equilateral triangles, squares or pentagons, it is useful to know the formulas for the circumscribed and inscribed circles for these plane figures as well as the size of the equal angles at each vertex.  The circumscribed circle is drawn about the figure, touching it at its vertices.  The inscribed circle is drawn within the figure and touches each of the sides at their midpoints.  Let rc and ri denote the radius of the circumscribed circle and inscribed circle respectively.

 

For any regular polygon of n sides, the size of the angle at each vertex is given by

 

Finding rc is equivalent to calculating the distance from a vertex to the centroid of a face.

Finding ri is equivalent to calculating the distance from a vertex to the mid-point of an edge.

Equilateral Triangle

 

For the equilateral triangle, the three sides are equal and denoted by a.  The three angles are also equal and are 60 degrees ().

 

        and     

 

Square

 

For the square, the four sides are equal and denoted by a.  The four angles are also equal and are 90 degrees ().

 

     and     

 

 

Pentagon

 

For the pentagon, the five sides are equal and denoted by a.  The five angles are also equal and are 108 degrees ().

 

 

        and     

 

 

Tetrahedron

 

A three-dimensional figure with four equilateral triangle faces, four vertices, and six edges.


   , where R is the radius of the circumscribed sphere.


r = (1/3)R,    where r is the radius of the inscribed sphere


 
 

Octahedron 

   

An octahedron is a three-dimensional figure with eight equilateral triangle faces, six vertices, and twelve edges. 

 

,  where R is the radius of the circumscribed sphere.

 

,  where R is the radius of the inscribed sphere.

 

Cube    

 

A cube is a three-dimensional figure with six square faces, eight vertices,  and twelve edges. 

 

,  where R is the radius of the circumscribed sphere.

 

 ;     ;    

 

 

 

Icosahedron 

 

 An icosahedron is a three-dimensional figure with twenty equilateral triangle faces, twelve vertices, and thirty edges. 

 

  , where R is the radius of the circumscribed sphere.

 

 

 

 

 

Dodecahedron 

    

A three-dimensional figure with twelve regular pentagon faces, twenty vertices, and thirty edges. 

 

  , where R is the radius of the circumscribed sphere.

 

 

 

 

 

r’ = 0.3205d   where d is the diameter of the circumscribed sphere

                        and r’ is the distance from a primary point to a constellation point.

 

 


PREVIOUS