According to Crofton's formula, the surface area $S(A)$ of a sufficiently regular compact set $A$ in $\mathbb{R}^d$ is proportional to the mean of all total projections $p_A \left( u \right)$ on a linear hyperplane with normal $u$, uniformly averaged over all unit vectors $u$. In applications, $p_A \left( u \right)$ is only measured in $k$ directions and the mean is approximated by a finite weighted sum $\widehat S \left( A \right)$ of the total projections in these directions. The choice of the weights depends on the selected quadrature rule. We define an associated zonotope $Z$ (depending only on the projection directions and the weights), and show that the relative error $\widehat S \left( A \right) / S \left( A \right)$ is bounded from below by the inradius of $Z$ and from above by the circumradius of $Z$. Applying a strengthened isoperimetric inequality due to Bonnesen, we show that the rectangular quadrature rule does not give the best possible error bounds for $d=2$. In addition, we derive asymptotic behavior of the error (with increasing $k$) in the planer case. The paper concludes with applications to surface area estimation in design-based digital stereology where we show that the weights due to Bonnesen's inequality are better than the usual weights based on the rectangular rule and almost optimal in the sense that the relative error of the surface area estimator is very close to the minimal error.

Key words: surface area, perimeter, Crofton formula, minimal annulus, isoperimetric inequality, associated zonotope, digitization.