I derived the Mean Magnetic length of toroidal core recently. By basic Inductance and Reluctance Concept coupled with few correct integrations, its simple to derive.
Needed(designed) Effective Mean Magnetic length for designing core of transformer depends on following data:
Normal hysteresis loop
Magnetization curve.
Total effective voltage to be impressed on transformer circuit.
Total resistance of circuit.
Total inductance of circuit.
For more detailed study to find mean magnetic length for given magnetizing current refer my research paper,' Simple Accurate Laboratory Method to find Magnetizing...........Including C.T effect.' Refer book' Electrical Machine Design' by A. K. Swahney.