TITLE: High-dimensional shapes with known volume formulas
QUESTION [8 upvotes]: There don't seem to be a lot of high-dimensional shapes whose volume, surface area, etc. can be expressed in a concise way.  The examples I know of are:

Spheres
Cubes (or parallelotopes, more generally)
Simplices
Zonotopes

What other classes of high dimensional objects admit relatively simple volume (or area, etc.) formulas?

EDIT: Since zonotopes are the most unfamiliar of my examples, here's a reference: Chapter 9 of "Computing the Continuous Discretely".  To summarize, a zonotope is a set of the form 
$$\{a_1 \vec{x}_1 + \cdots + a_m \vec{x}_m \:|\: a_1,\dots,a_m\in[0,1]\}$$
where $\vec{x}_1,\dots,\vec{x}_m\in \mathbb{R}^n$ are fixed.  This is like a parallelotope, except the vectors $\vec{x}_j$ need not be linearly independent (e.g. $m$ can be greater than $n$).  The volume of such a zonotope is given by 
$$ \sum_{S\subset \{1,\dots,m\}, |S|=n} |\det[x_i]_{i\in S}|$$
which means: "Take any n of the m vectors $\vec{x_i}$ and compute the volume of the parallelotope formed by these n vectors in $\mathbb{R}^n$.  Sum over all such parallelotopes and you get the volume of the zonotope."

REPLY [3 votes]: A couple more examples I've found:
1) Cross-polytopes.  These are generalizations of an octohedron.  Wikipedia has a nice article on them, and the standard n-dimensional cross-polytope has volume $\frac{2^n}{n!}$.
2) Cones.  A cone is formed from a base shape of codimension 1 and a point at some height $h$ above the base.  In n dimensions, if the base has $(n-1)$ volume $A$, then the cone has $n$ volume $\frac{Ah}{n}$.