We demonstrate that when a single-walled carbon nanotube is under pressure it undergoes a series of shape transitions, first transforming from a circle to an oval and then from an oval to a peanut. Most remarkably, the ratio of the area of the tube cross sections at the second transition over that at the first transition appears as a constant, independent of the tube radius. Its accurate value is computed to be G=0.819 469, by formulating a variational geometry problem to represent single-walled carbon nanotubes with a family of closed plane curves of fixed length and minimum bending energy. The implications of such a geometric constant in designing nanotube electromechanical pressure sensors are discussed.