  We deduce explicit formulae for the intrinsic volumes of an ellipsoid in
$\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an
ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we
show that
  \begin{align*} V_k({\mathcal
E})=\kappa_k\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\,\rm{d}t
\end{align*} for all $k=1,\ldots,d$, where $s_{k-1}$ is the $(k-1)$-th
elementary symmetric polynomial and $\kappa_k$ is the volume of the
$k$-dimensional unit ball. Some examples of the intrinsic volumes $V_k$ with
low and high $k$ are given where our formulae look particularly simple. As an
application we derive new formulae for the expected $k$-dimensional volume of
random $k$-simplex in an ellipsoid and random Gaussian $k$-simplex.
