The Tetrahedron

 

The tetrahedron is a solid figure having four faces each of which is an equilateral triangle.  It has four vertices and six edges.

 

 

Refer to Figure 6.1 A, B and C.  In the figure the following notation is used:

 

a          is the length of a side

h          is the height of the figure

r           is the radius of the inscribed sphere

R         is the radius of the circumscribed sphere

 

Q

is the angle that a line from a vertex to the centroid of the figure from a face

x         

is the distance from a vertex to the centroid of the ‘base’.  It is the radius of the circumscribing circle for the equilateral triangle that is the bottom of the figure.  See Figure 6.1B.

 

From Equation 3.2) above, we know that

 

 

Figure 6.1.B shows the ‘bottom’ of the tetrahedron.  The distance marked d is the distance from the midpoint of a face to the centroid of the face.  That is, d is the radius of the circle inscribed in the equilateral triangle that is the bottom of the figure.

 

 

 

 

To find the height of the figure, h,  consider Figure 6.1.C.

 

 

so we have that

 

h = a  =  a = 

 

So finally we have

 

6.1)      h = 

 

To find the radius,R, of the circumscribed sphere, consider Figure 7.1.C

 

 

 

And from the figure we see that r = h – R  so that we have

 

 

 

By collecting and simplifying, we have the following

 

 

 

So  2Rh =

 

or

 

R =   =    =  

 

R =

 

So finally we have that

 

6.2)      R =        where R is the radius of the circumscribing sphere

 

 

To find the radius of the inscribed sphere, note from Figure 6.1.C that r = h –R.

 

This gives the relationship

 

r =

 

6.3)      r =   where r is the radius of the inscribed sphere.

To express r in terms of R, where R =

we have

 

6.4)      r =

To express a in terms of R,  we have a =   and we have finally

6.5)      a =

 

 

Figure 6.1

 

Reference [5]