The relative response to X-rays and light, linearity for Xrays, and the energy resolution for both X-rays and light signal from a light emitting diode (LED) light pulser were measured for 5 mm, 10 mm and 16 mm diameter Large Area Avalanche Photodiodes (LAAPDs). It has been shown that the ratio of the LAAPD gain for 5.9 keV and 16.6 keV X-rays and the gain for a light pulse from the LED light pulser was constant (within 1-2%) up to a device (optical) gain of 100. That allows using the X-ray peaks as a good reference for measuring the e-h pair number of detected light pulses, particularly from scintillators. Good linearity of the LAAPDs response to X-rays was observed up to energy of about 20 keV. The energy resolutions for light signals and X-rays were compared and discussed in the light of the statistical limitation of the resolution. The comparison showed an excellent agreement for light signal. For the X-rays, the energy resolution was affected by the non-uniformity of the gain in the active area of the photodiode, particularly observed for X-rays above 6 keV. Finally, the energy spectrum fur 662 keV gamma-rays from a Cs-137 source detected with a YAP crystal coupled to a 16 mm LAAPD was obtained, showing the excellent energy resolution of 4.30+/-0.15%.