Thin components made from advanced brittle glasses or ceramics are becoming increasingly important due to the widespread adoption of portable consumer products as well as other modern electronic and medical devices. The strength of these brittle materials is traditionally estimated from empirical relationships relating the stress at failure to characteristic lengths derived from the fracture surface's topography. One example is Orr's relationship, sigma(f).R-m(1/2)=A(m), which correlates the material strength, sigma(f), to the radius of the mirror-mist boundary region, R-m, through the empirical constant, A(m). Although various studies have shown that, for flexural fractures (failed in bending), A(m) depends on the specimen's geometry, this effect has been generally neglected by arguing that the magnitude of A(m) is almost constant for thicker specimens. However, we show that this argument cannot be applied to thin geometries, and that by not accounting for the thickness of the sample, the flexural strength will be grossly underestimated. In this work, we introduce an expression based on an iterative fracture mechanics algorithm which yields more accurate estimates of flexural strength for thin brittle components in bending. The accuracy of the model is validated both through flexural strength tests on glass and by comparing our predictions to an extensive literature survey of experimental results.