The multifractal scaling of velocity increments at small scales poses significant challenges to turbulence theories based on structure functions of velocity increments. Velocity circulation has recently emerged as a physically pertinent quantity with simpler inertial-range statistics. It therefore offers a potential route to simplified descriptions of turbulent intermittency. Here, we assess the applicability of circulation-based scaling theories in free-surface turbulence using planar particle image velocimetry below a quasi-flat interface. We show that the circulation around closed contours obeys the area rule in this flow. When all loop side lengths lie within the same scaling range, the circulation statistics depend primarily on the enclosed area rather than on loop shape or aspect ratio. For figure-eight contours, the relevant area is the scalar enclosed area rather than the signed algebraic area. We further find that inertial-range circulation moments exhibit bifractal scaling. Low-order moments are consistent with space-filling behaviour, whereas high-order moments follow a monofractal branch with an effective dimension close to 2.2. These results indicate that circulation provides a compact description of intermittency in free-surface turbulence, despite the mixed two- and three-dimensional character of the flow.