Area–volume properties of fluid interfaces are investigated to quantify the scale-local and cumulative structure. An area–volume density g3(λ) and ratio Ω3(λ) are introduced to examine the interfacial behaviour as a function of scale λ or across a range of scales, respectively. These measures are demonstrated on mixed-fluid interfaces from whole-field ∼10003 three-dimensional space–time concentration measurements in turbulent jets above the mixing transition, at Re ∼ 20000 and Sc ∼ 2000, recorded by laser-induced-fluorescence and digital-imaging techniques, with Taylor's hypothesis applied. The cumulative structure is scale dependent in Ω3(λ), with a dimension D3(λ) that increases with increasing scale. In contrast, the scale-local structure exhibits self-similarity in g3(λ) with an exponent αg ≈1.3 for these interfaces. The scale dependence in the cumulative structure arises from the large scales, while the self-similarity corresponds to the small-scale area–volume contributions. The small scales exhibit the largest area–volume density and provide the dominant contributions to the total area–volume ratio, which corresponds to ∼10 times the area of a purely large-scale interface for the present flow conditions. The self-similarity in the scale-local structure at small scales provides the key ingredient to extrapolate the area–volume behaviour to higher Reynolds numbers.