The paper presents the design principles of a degenerate mode resonant mass sensor in which the unloaded sensor takes the form of a cyclically symmetric structure. The simplest structure with these features is a circular diaphragm and its properties are exploited in this paper. Such structures support pairs of independent modes of vibration which share a common natural frequency and these are referred to as degenerate modes. If extra mass is added to the structure over predefined regions, then the degeneracy can be broken and this produces a separation of the previously identical frequencies. This frequency split is the output of the sensor and is proportional to the added mass. Such a sensor is self compensating and ambient effects which equally influence both modes, such as temperature and in-plane stress, do not add to the frequency split. A Lagrangian approach is used to derive the relationship between added mass and frequency split.