The aim of the paper is to solve the problem: to reconstruct integrals of a function over \(p\)-dimensional planes in \(\mathbb{R}^n\) starting from its integrals over \(k\)-planes, where \(p< k\). It is proved that for every \(p<k\) there exists an operator \(K_{PK}\) from the space of functions on the manifold \(H_K\) of all \(k\)-dimensional planes into the space of differential \((k-p)\)-forms on \(H_K\) with specific properties. The intermediate inversion formulas are generalized to integral transforms from functions on \(H_P\) to functions on \(H_K\), where \(0\leq p<k\).