  The Grassmann angle improves upon similar angles between subspaces that
measure volume contraction in orthogonal projections. It works in real or
complex spaces, with important differences, and is asymmetric, what makes it
more efficient when dimensions are distinct. It can be seen as an angle in
Grassmann algebra, being related to its products and those of Clifford algebra,
and gives the Fubini-Study metric on Grassmannians, an asymmetric metric on the
full Grassmannian, and Hausdorff distances between full sub-Grassmannians. We
give formulas for computing it in arbitrary bases, and identities for angles
with certain families of subspaces, some of which are linked to real and
complex Pythagorean theorems for volumes and quantum probabilities. Unusual
features of the angle with an orthogonal complement, or the angle in complex
spaces, are examined.
