Tetrahedron Surface Area
Python, 198 178 161 chars

V=eval('input(),'*4)
A=0
for i in range(4):F=V[:i]+V[i+1:];a,b,c=map(lambda e:sum((a-b)**2for a,b in zip(*e)),zip(F,F[1:]+F));A+=(4*a*b-(a+b-c)**2)**.5
print A/4


The input format is as given in the question.

It calculates the length of the edges adjacent to each of the faces and then uses Heron's formula.