It is shown that the well-known linear variation of p-n diode terminal voltage with temperature at different fixed forward currents allows easy and accurate determination of the semiconductor ideality factor and bandgap from only two data points. This is possible if the temperature difference required to maintain the same diode voltage drop can be measured. The results for silicon and germanium bandgap energy using this approach are in excellent agreement with the literature. The method therefore provides a fresh and original insight into the derivation of the bandgap using data from a popular experiment. The experiment is suitable for undergraduate laboratories on semiconductors. (Contains 2 tables and 3 figures.)