In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set E := { z : | z | ≤ r E:=\big \{z\,:\, |z|\le r and | P ( z ) | ≤ ϵ n } |P(z)|\le \epsilon ^{n}\big \} , for a polynomial P P of degree ≤ n \le n . Usually, P P is taken to be monic, and either Cartan’s Lemma or potential theory is used to estimate the size of E E , in terms of Hausdorff contents, planar Lebesgue measure m 2 m_{2} , or logarithmic capacity cap. Here we normalize ‖ P ‖ L ∞ ( | z | ≤ r ) = 1 \|P\|_{L_{\infty }\bigl (|z|\le r\bigr )}=1 and show that cap ( E ) ≤ 2 r ϵ (E)\le 2r\epsilon and m 2 ( E ) ≤ π ( 2 r ϵ ) 2 m_{2} (E)\le \pi (2r\epsilon )^{2} are the sharp estimates for the size of E E . Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on C n \mathbb {C}^{n} or product capacity and Favarov’s capacity. Several of our estimates are sharp with respect to order in r r and ϵ \epsilon .