Hint: As a numerical analyst, the most common way to compute the face are of a bunch of tetrahedra is using cross product, since it is the most easily vectorized/paralleled algorithm. The face opposite to $A$ denoted as $F_A$ is spanned by the vector $\overrightarrow{BC}$ and $\overrightarrow{BD}$, then the area is $ |F_A| = \frac{1}{2} |\overrightarrow{BC}\times \overrightarrow{BD}| $
More likely since you mentioned "input", I am guessing you are writing some subroutine, if using cyclic notation that a tetrahedron has vertices $V_i$, $i=1,\cdots,4$, then above formula can be vectorized using the following MATLAB code snippets assuming you have your $i$-th vertex of the $n$-th tetrahedron stored in a 3d-array V(n,:,i)
:
face_normal = cross(V(:,:,i+1) - V(:,:,i-1), ... V(:,:,i+1) - V(:,:,i+2), 2); face_area = 0.5*sqrt(sum(face_normal.^2,2));
which could be easily ported to Python, C or Fortran.