  Let $K$ be a $d$ dimensional convex body with a twice continuously
differentiable boundary and everywhere positive Gauss-Kronecker curvature.
Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently
from $K$ according to the uniform distribution. Matching lower and upper bounds
are obtained for the orders of magnitude of the variances of the $s$-th
intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore,
strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The
essential tools are the Economic Cap Covering Theorem of B\'ar\'any and Larman,
and the Efron-Stein jackknife inequality.
