Given a closed polygon P having n edges, embedded in R^d, we give upper and lower
bounds for the minimal number of triangles t needed to form a triangulated PL surface in
R^d having P as its geometric boundary. The most interesting case is dimension 3, where the
polygon may be knotted. We use the Seifert suface construction to show there always exists
an embedded surface requiring at most 7n^2 triangles. We complement this result by showing
there are polygons in R^3 for which any embedded surface requires at least 1/2n^2 - O(n)
triangles. In dimension 2 only n-2 triangles are needed, and in dimensions 5 or more there
exists an embedded surface requiring at most n triangles. In dimension 4 we obtain a
partial answer, with an O(n^2) upper bound for embedded surfaces, and a construction of an
immersed disk requiring at most 3n triangles. These results can be interpreted as giving
qualitiative discrete analogues of the isoperimetric inequality for piecewise linear
manifolds.