A certain approximation scheme based on "piecewise linear" approximations of L2spaces is employed to formulate a numerical method for solving quadratic optimal control problems governed by linear retarded functional differential equations. This piecewise linear method is an extension of the so-called averaging technique. It is shown that the Riccati equation for the linear approximation is solved by simple transformation of the averaging solution. In fact, the numerical procedures are identical, except for the computation of an initial condition. Numerical examples are included.