Thanks to their large angular dimension and brightness, red giants and supergiants are privileged targets for optical long-baseline interferometers. 16 red giants and supergiants have been observed with the VLTI/AMBER facility over a 2-year period, at medium spectral resolution ( ) in the K band. The limb-darkened angular diameters are derived from fits of stellar atmospheric models on the visibility and the triple product data. The angular diameters do not show any significant temporal variation, except for one target: TX Psc, which shows a variation of 4 per cent using visibility data. For the eight targets previously measured by long-baseline interferometry (LBI) in the same spectral range, the difference between our diameters and the literature values is less than 5 per cent, except for TX Psc, which shows a difference of 11 per cent. For the eight other targets, the present angular diameters are the first measured from LBI. Angular diameters are then used to determine several fundamental stellar parameters, and to locate these targets in the Hertzsprung-Russell diagram (HRD). Except for the enigmatic Tc-poor low-mass carbon star W Ori, the location of Tc-rich stars in the HRD matches remarkably well the thermally-pulsating asymptotic giant branch, as it is predicted by the stellar evolution models. For pulsating stars with periods available, we compute the pulsation constant and locate the stars along the various sequences in the period-luminosity diagram. We confirm the increase in mass along the pulsation sequences, as predicted by theory, except for W Ori which, despite being less massive, appears to have a longer period than T Cet along the first-overtone sequence.