Is there any natural generalization of pythagoras theorem to higher dimensions?

From http://www.cs.bc.edu/~alvarez/NDPyt.pdf :

... we will restrict our attention to the analogs
of right triangles. We will call these orthogonal tetrahedrons.
An orthogonal tetrahedron is any tetrahedron which has a vertex
at which three faces meet at right angles to each other ...
we refer to [these faces] as the orthogonal faces,
and we refer to the remaining face as the opposing face ...

Let $A$ be the area of the opposing face, and let
$A_1 , A_2 , A_3$ be the areas of the
orthogonal faces of a given orthogonal tetrahedron.
Then the following relation holds:
$A^2 = A_1^2 + A_2^2 + A_3^2$

Next we address the n-dimensional case ...

Let $A$ be the area of the opposing face, and let
$A_1, A_2, ..., A_n$ be the areas of the orthogonal faces
of a given orthogonal n-simplex. Then the following relation holds:

$$A^2 = \sum_{j=1}^n A_j^2$$