A solution is given to the problem of where to choose $k + 1$ points and what weights to assign them in order to obtain the minimum variance of what may be called an interior extrapolated value of a polynomial of degree $k$. It is assumed that observations can be taken in the interval $\lbrack -1, 1\rbrack$, except for a subinterval $(\alpha, \beta)$ located in its interior and within which the extrapolation occurs. This one dimensional solution is then used to solve the corresponding two dimensional problem, for a certain class of polynomials, in which it is desired to extrapolate inside a rectangular region that lies inside a larger rectangular region and within which observations can be taken, exclusive of the interior region. In addition, the two dimensional exterior extrapolation problem is solved for the same class of polynomials as those used for interior extrapolation.