The closed form of a rotational version of the famous Crofton formula is derived. In the simplest case where the sectioned object is a compact subset of $\mathbb{R}^d$ with a $(d-1)$-dimensional manifold of class $C^2$ as boundary, the rotational average of intrinsic volumes measured on sections passing through a fixed point can be expressed as an integral over the boundary involving hypergeometric functions. In the more general case of a compact subset of $\mathbb{R}^d$ of positive reach, the rotational average also involves hypergeometric functions.