We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The control is the trace of the state on the boundary of the domain, which is assumed to be a convex, polygonal, open set in R-2. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the error estimates are of order O(h(1-1/p)) for some p > 2, which is consistent with the W-1-1/p,W-p(Gamma)-regularity of the optimal control.