The pursuit of low noise in optical instruments for areal surface topography measurement is relevant to many surface types, ranging from super-polished optical surfaces to weakly reflecting or scattering textures that require enhanced signal sensitivity. We clarify the definition and experimental methods for quantifying random noise in areal surface topography measurements. We also propose a parameter, the topographical noise density, that concisely summarizes the effects of measurement bandwidth. To illustrate these ideas, we present results from a commercial phase-shifting interference microscope showing an RMS measurement noise of 0.03 nm for a 1-s data acquisition of 1 million surface topography image points, after application of a 3  ×  3-pixel convolution filter. The results follow the expected inverse square root dependence on the data acquisition time for fast averaging of topography maps, resulting in a measurement noise of <0.01  nm for a 10-s data acquisition.