Using the classical power function relationship between inverse plot size and sample variance, the method of Lagrangian multipliers was applied to minimize the variance of the sample mean among measurements on sample plots (e.g. volume or basal area per hectare) subject to a fixed total cost with varying plot size and sample size. This results in equations that can be used to determine optimal plot and sample size for a fixed total cost. The method does not require a prior estimate of variance. Optimal point sampling basal area factor can be substituted for plot size if variance and cost functions are available for point sampling. It was determined theoretically that no feasible optimum solution exists if the ratio of the relative change rate in variance to the relative change rate in cost per plot is less than negative one. This occurs because in that case the variance as a function of plot size is being reduced faster than the cost per plot is increasing in relative terms. An example application based on cost and variance functions from the literature found that the optimal surface was quite flat, so that an optimum existed but a rather wide range of plot and sample size combinations were close to optimum for a fixed total cost. In practical terms this means that a wide range of plot sizes would give nearly equal precision for a fixed total cost in this example. This may not be the case for all realistic parameterizations of the cost functions.