A rotational version of the famous Crofton formula is derived. The motivation for deriving the formula comes from local stereology, a new branch of stereology based on sections through fixed reference points. The formula shows how rotational averages of intrinsic volumes measured on sections passing through fixed points are related to the geometry of the sectioned object. In particular it is shown how certain weighting factors, appearing in the rotational integral formula, can be expressed in terms of hypergeometric functions. Close connections to geometric tomography will be pointed out. Applications to stereological particle analysis are discussed.