Two polytopes in Euclidean n-space are called scissors congruent if one can be cut into finitely many polytopic pieces that can be rearranged by Euclidean isometries to form the other. A generalized version of Hilbert's third problem asks for a classification of Euclidean n-polytopes up to scissors congruence. A homeomorphism of the Cantor set is contained in the topological full group of a Cantor dynamical system if the Cantor set can be cut into finitely many clopen pieces on which the homeomorphism is described by the dynamics. Conjectures of Matui predict a relation between this group, an ample groupoid encoding the dynamics and the K-theory of a related C*-algebra. In this talk, I will use the group of interval exchange transformations as an example to outline connections between these two subjects. This is based on joint work with Kupers--Lemann--Malkiewich--Miller, which relates Zakharevich's higher scissors congruence K-theory and calculations obtained by Malkiewich to recent advances of Li and Tanner on Matui's conjectures and a result of Szymik--Wahl proving that Thompson's group V is acyclic.