The paper deals with an inversion formula for the \(k\)-plane Radon transform \(R^kf\) in a space of constant curvature. Both euclidean and noneuclidean cases are considered. An explicit formula from \(R^kf\) to \(R^{k-2}f\) is obtained. Techniques of geometrical integration and the formula for the second derivation of \(f\circ\gamma\) \((\gamma\) is a regular parametrized smooth curve in \(X_k^n\), \(X^n_k\) is the \(n\)-dimensional real standard space of constant curvature) are used. An application of the reduction step leads to the classical inversion formula.