Cell invasion through a dense three-dimensional (3D) matrix is believed to depend on the ability of cells to generate traction forces. To quantify the role of cell tractions during invasion in 3D, we present a technique to measure the elastic strain energy stored in the matrix due to traction-induced deformations. The matrix deformations around a cell were measured by tracking the 3D positions of fluorescent beads tightly embedded in the matrix. The bead positions served as nodes for a finite element tessellation. From the strain in each element and the known matrix elasticity, we computed the local strain energy in the matrix surrounding the cell. We applied the technique to measure the strain energy of highly invasive MDA-MB-231 breast carcinoma and A-125 lung carcinoma cells in collagen gels. The results were compared to the strain energy generated by non-invasive MCF-7 breast and A-549 lung carcinoma cells. In all cases, cells locally contracted the matrix. Invasive breast and lung carcinoma cells showed a significantly higher contractility compared to non-invasive cells. Higher contractility, however, was not universally associated with higher invasiveness. For instance, non-invasive A-431 vulva carcinoma cells were the most contractile cells among all cell lines tested. As a universal feature, however, we found that invasive cells assumed an elongated spindle-like morphology as opposed to a more spherical shape of non-invasive cells. Accordingly, the distribution of strain energy density around invasive cells followed patterns of increased complexity and anisotropy. These results suggest that not so much the magnitude of traction generation but their directionality is important for cancer cell invasion.