Quantifying blood flow is of major clinical importance for the assessment of the cardiovascular status. Longitudinal flow measurements in the common carotid suffer from operator-dependency and are very sensitive to motion. In this work, a method is proposed to perform flow estimation from cross-sectional acquisitions, which reduces operator-dependency and is more robust to motion. By modeling the vessel as a cylinder, the intersection between the ultrasound plane and the vessel is an ellipse. The properties of this ellipse (semi-major and semi- minor axis, rotation and center position) are used to estimate the Doppler angle (beam-to-flow angle). This method was tested in vitro on a constant flow phantom using a wide variety of Doppler angles, where the errors in the flow estimates were below 10%. Further research should aim at quantifying the sensitivity of this method and validating this method in vivo.