Learning games have great potential to become an integral part of new classrooms of the future. One of the key reported benefits is the capacity to keep students deeply engaged during their learning process. Therefore, it is necessary to develop models that can measure quantitatively how learners are engaging with learning games to inform game designers and educators, and to find ways to maximize learner engagement. In this article, we present our proposal to multidimensionally measure engagement in a learning game over four dimensions: general activity, social, exploration, and quests. We apply metrics from these dimensions to data from The Radix Endeavor, an inquiry-based online game for STEM learning that has been tested in K-12 classrooms as part of a pilot study across numerous schools. Based on these dimensions, we apply clustering and report four different engagement profiles that we define as "integrally engaged," "lone achiever," "social explorer," and "nonengaged." We also use three variables (account type, class grade, and gender) to perform a cross-sectional analysis finding interesting, statistically significant differences in engagement. For example, in-school students and accounts registered to males engaged socially much more than out-of-school learners or accounts registered to females, and that older students have better performance metrics than younger ones.