He could have measured the angle of the desk's diagonal makes with the longer side, that would give him tan(w/l), no inches or centimetres required. You do nee

He could have measured the angle of the desk's diagonal makes with the longer side, that would give him tan(w/l), no inches or centimetres required. You do need radians, but that's not a unit in the same way a 'meter' is, you don't need any kind of reference to know how much, say, 1 pi radians is. For most intents and purposes angles are unitless.