The surface area of an object revisited—but from random directions

The estimation of surface areas visualized by macroscopy, by light microscopy or at the ultrastructural level is important in several areas of biological research. Essentially, there are two empirical approaches to surface estimation. The first, the model-based approach, depends on modelling objects by geometric shapes whose surface area can be calculated from mathematical expressions (e.g. modelling apples and avian erythrocytes as ellipsoids). This method is not unbiased in general and, in many cases, the magnitude and direction of the bias are not known. The second, design-based , approach is much better. It is unbiased in general (so applies to arbitrary objects) and is efficient. It requires isotropic sectioning of real objects in three- or two-dimensional space. The design-based approach is illustrated by application to apples and to fixed human brains.