Consider a random smooth Gaussian field G(x): F→ ℝ, where F is a compact set in ℝd. We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles. © 2011 Springer Science+Business Media, Inc.