The area Σ of a parallelogram with sides "A" and "B" containing the angle θ is:
which will be recognized as the magnitude of the vector cross product of the vectors A and B lying along the sides of the parallelogram. That is:
The square of this expression is:
where Γ(A, B) is the Gram determinant of A and B defined by:

In a similar fashion, the squared volume "V" of a parallelepiped spanned by the three vectors A, B and C is given by the Gram determinant of the three vectors:
This process can be extended to "n"-dimensions.

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