The Octahedron

 

 

 

 

An octahedron is a solid figure having 8 sides, 6 vertices, 12 edges and 8 faces.

Each face is an equilateral triangle.  The ‘base’ of an octahedron is a square.

 

To find the radius of the circumscribing sphere, we need only to compute the distance from the center of the square base to a vertex of the square.

 

Figure 7.1 shows isosceles triangles formed in half of the square base.  The distance from the center of the square to a vertex of the square is the length of the radius, R, of the circumscribing sphere.

 

We know from Equation 4.2) that

 

7.1)      R =  

 

where a is the length of a side and R is the radius of the circumscribed sphere.

 

 

Figure 7.2 shows one face of an octahedron whose sides are equal to a.

From this figure, the slant height of the face, h, can be calculated as follows:

 

 

So,

 

h =

 

Continuing with Figure 7.2, the distance along the slant height to the centroid of the triangle, h₁, can be calculated.

 

 

So we finally have

 

 

 

In Figure 7.3 we have a triangle formed in a plane that is through the 'top' vertex of the octahedron, perpendicular to the square base of the figure, and intersecting one side at its midpoint.

 

h is the slant height just calculated.  h₁ is the distance along the slant height to the centroid of a face.  r is the radius of the inscribed sphere that touches each face at its centroid.

 

From this figure, we can see that

 

 

So then

 

 

 

so 

 

And finally

7.2)     

 

It is interesting to calculate the angles a and b.

 

From Figure 7.3 we have the following:

 

 

b = 90 – a = 35.26 degrees

 

 

 

To find a and r in terms of R we start with Equation 7.1.  This equation gives

 

 

 

So, solving for a we get

 

 

 

         But we also know that

 

So we have

 

 

So finally we have

 

7.3)                 

 

where a is the length of a side and R is the radius of the circumscribed sphere.

 

 

To find r in terms of R consider Equation 7.2:

 

                   Substituting the value of a in Equation 7.3 here gives

 

 

 

And so finally we have

 

7.4)                 

 

where r is the radius of the inscribed sphere and R is the radius of the circumscribed sphere.

 

 

 

 

Figure 7.1

 

Isosceles triangles formed in the square base of an octahedron.

 

 

 

 

Figure 7.2

 

Equilateral triangle that is one face of an octahedron.

 

 

 

 

 

Figure 7.3

 

Right triangle in the plane through the top vertex of the octahedron and through the midpoint of one of the sides.