  Thermodynamic properties of confined systems depend on sizes of the
confinement domain due to quantum nature of particles. Here we show that shape
also enters as a control parameter on thermodynamic state functions. By
considering specially designed confinement domains, we separate the influences
of quantum size and shape effects from each other and demonstrate how shape
effects alone modify Helmholtz free energy, entropy and internal energy of a
confined system. We propose an overlapped quantum boundary layer method to
analytically predict quantum shape effects without even solving Schr\"odinger
equation or invoking any other mathematical tools. Thereby we reduce a
thermodynamic problem into a simple geometric one and reveal the profound link
between geometry and thermodynamics. We report also a torque due to quantum
shape effects. Furthermore, we introduce isoformal, shape preserving, process
which opens the possibility of a new generation of thermodynamic cycles
operating at nanoscale with unique features.
