Mesoscale eddies are ubiquitous in the global ocean. Most of them are materially coherent: they advect a different water mass in their core than in the surrounding water, according to studies based on in situ observations and Lagrangian techniques. In parallel, laboratory experiments have shown that eddies have the ability to locally modify the stratification according to the thermal wind balance, without necessarily contain heterogeneous water. These two types of density anomalies associated with mesoscale eddies are often erroneously confused in the literature. Here we propose a new theoretical decomposition of the potential density field in the core of eddies to assess their respective amplitude and dynamical effect. This allows the modelling of their 3D shape and the estimation of the importance of each term. This decomposition is applied to 6 anticyclonic eddies sampled during the EUREC4A-OA, METEOR 124 and PHYSINDIEN 2011 in situ experiments. We show that the anomaly corresponding to the slope of the isopycnals is the largest contributor to the total density anomaly. Its vertical shape is nearly Gaussian, but also depends on the local background stratification. The horizontal density gradient associated with the trapped water mass adds a second order term to the total anomaly and can be neglected for the study of eddy dynamics. The horizontal structures of the eddies studied are consistent with previous studies and show an exponential-alpha shape.