We use the cut and paste relation [Y [2] ] = [Y ][P m ] + L 2 [F (Y )] in K 0 (Var k ) of Galkin-Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. The same method also gives a family of other cut and paste relations that can only possibly be satisfied by cubic hypersurfaces.Suppose that Y is not a 1-dimensional family of quadrics or a 2-dimensional family of (projective) (m − 2)-planes (each isomorphic to P m−2 ). If m ≥ 7, then the following are equivalent:2. Y is a cubic hypersurface or the intersection of two quadric hypersurfaces. This statement can be replaced with Y being a cubic hypersurface if we assume that F (Y ) is connected. 3 (p. 12 of [16]).Remark 0.3. Here are some comments on the assumptions of Theorem 0.2. 1. If Hartshorne's conjecture (part 1 of Remark 1.7) holds in codimension 2, then the conditions on Y can be weakened to 2d − 4 ≤ n and n ≥ 7. Note that the uniruledness property already implies d ≤ n if Y is a hypersurface and d 1 + d 2 ≤ n if Y has codimension 2 and d 1 , d 2 are the degrees of the hypersurfaces whose intersection is equal to Y . Another possible replacement of the conditions is to take Y contained in some hypersurface of degree d such that 2d − 4 ≤ n.