In this paper, we study how well one can approximate arbitrary polytopes using sparse inequalities. Our motivation comes from the use of sparse cutting-planes in mixed-integer programing (MIP) solvers, since they help in solving the linear programs encountered during branch-&-bound more efficiently. However, how well can we approximate the integer hull by just using sparse cutting-planes? In order to understand this question better, given a polyope $$P$$P (e.g. the integer hull of a MIP), let $$P^k$$Pk be its best approximation using cuts with at most k non-zero coefficients. We consider $${\text {d}}(P, P^k) = \max _{x \in P^k} \left( \min _{y \in P} \Vert x - y\Vert \right) $$d(P,Pk)=maxx?Pkminy?P?x-y? as a measure of the quality of sparse cuts.In our first result, we present general upper bounds on $${\text {d}}(P, P^k)$$d(P,Pk) which depend on the number of vertices in the polytope. Our bounds imply that if $$P$$P has polynomially many vertices, using half sparsity already approximates it very well. Second, we present a lower bound on $${\text {d}}(P, P^k)$$d(P,Pk) for random polytopes that show that the upper bounds are quite tight. Third, we show that for a class of hard packing IPs, sparse cutting-planes do not approximate the integer hull well, that is $$d(P, P^k)$$d(P,Pk) is large for such instances unless k is very close to n. Finally, we show that using sparse cutting-planes in extended formulations is at least as good as using them in the original polyhedron, and give an example where the former is actually much better.