Volume of a regular tetrahedron

At the end of the day when I was working with Justin on Math, John came and asked me what is the volume of a tetrahedron having all sides equal to 15 cm. I figured that it would benefit everyone to share the solution, and so here it is.

Referring to the accompanying diagram, and assuming that the length of the sides is c:
let ABCD be the tetrahedron (a pyramide with a triangular base),
M be the mid-point of AB, thus CM is the median of triangle ABC.
O be the centroid of the triangle ABC, having CO=2 x OM. Note that OA bisects angle BAC into 2x30 degrees. Consider triangle MAO, OA = 2 OM = OC.
We will proceed to find the area of the base, ABC:
Area of the base is Base x Height /2 = MB x MC
MB = AB/2 = 0.5c
Using Pythagoras theorem on triangle MBC,
MC2 = BC2-MB2 = c2-(0.5c)2 = 0.75c2
MC=sqrt(3)c/2
Therefore,
area of the base
= MB x MC = 0.5c x sqrt(0.75) c
= sqrt(3)c2/4

Now let's work on the height, h, of the tetrahedron:
h=OD
=sqrt(DM2-OM2)
=sqrt(MC2-(MC/3)2)
=2sqrt(2)MC/3
=sqrt(6)c/3

Therefore the volume of the tetrahedron is:
Volume
= Area of base x height /3
=sqrt(3)c2/4 sqrt(6)c/3 /3
= sqrt(2)c3/12

For a side of 15 cm, we obtain
Volume = sqrt(2)(15)3/12=397.748 cm3

Volume of an arbitrary tetrahedron

It is easy to calculate the volume of a tetrahedron when the side lengths are equal using the above formula. What if we have a tetrahedron with unequal sides, in fact, all six sides are different?
Do we have a formula other than the classical base area x height/3?
Indeed! A formula exists that is based on the edge lengths of the tetrahedron, and they can be all different!
Let's first define the opposite edges of a tetrahedron as the edges which do not share a common vertex. In a tetrahedron, there are three such pairs.
If we denote the lengths of each pair by (u,U), (v,V) and (w,W) in any order, the volume of the tetrahedron is then:
Volume=sqrt(4u2v2w2 - u2(v2+w2-U2)2 - v2(w2+u2-V2)2 - w2(u2+v2-W2)2 + (v2+w2-U2)(w2+u2-V2)(u2+v2-W2))/12

If the above formula looks complicated, it is quite true. However, it is symmetrical, in that if u, v, w are interchanged, the formula will remain as it is. We can proceed to validate the formula by applying it to a regular tetrahedron. Substitute the lengths of all six sides (u, v, w, U, V, W) by the length c, and we get back the formula Volume=sqrt(2)c3/12

In fact, the formula is a development of a much simpler formula involving determinants, which is as follows:

Let the five rows of determinant D be defined as follows:
(0, u2, v2, w2, 1)
(u2, 0, W2, V2, 1)
(v2, W2, 0, U2, 1)
(w2, V2, U2, 0, 1)
(1,1,1,1,0)

The same volume can be simply obtained by:
Volume=sqrt(D/2)/12

References:
1. http://www.cs.berkeley.edu/~wkahan/VtetLang.pdf
2. Analytical Geometry of Three Dimensions, D.M.Y. Sommerville, Cambridge University Press 1951.