A smooth and bounded interpolant can be constructed in explicit algebraic form within any polygon, convex or concave. The resulting function is not unique and accordingly can be adjusted to satisfy desired global conditions, such as linear fields. The closed-form representation is obtained by combining simple geometric descriptions, such as the side lengths and areas. The interpolant distributes values given at discrete nodes smoothly over the interior of the domain. On a convex polygon, the interpolant is a rational function of the product of areas. On a concave or multiply connected polygon, the interpolant is a function of areas and edge lengths, which introduces a square root term.