The apathy radius is, quite simply, the distance which you can be bothered to travel for something.
Behind this simple truth, however, lies a complex mathematical model which has to take into account such diverse factors as temperature, time of day, the desirability of the something in question and how much you drank last night.
The calculus of apathy is in fact so insanely complex that nobody can be a***d to work it out. The closest we have yet come is to define certain localised phenomena, mainly to do with beds.
The Equiapathetic Surface
A duvet, it has been observed, is an equiapathetic surface. You can move freely under it without apparent effort. But try to stick more than a tiny proportion of your body outside the duvet and what happens? You hit the Alpine slopes of an apathy radius limit.
The Apathetic Tunnelling Effect
To observe this, drink six pints or more of beer and go to bed. In the morning1 you will wake up in urgent need of communion with the gods of the watercloset.
You sprint for the loo, past objects which you just can't be bothered to attend to, such as the telephone which has been ringing unheeded for the last half hour; the toaster which is quietly filling the house with smoke after your flatmate passed out waiting for it to pop up; and the scantily clad and highly attractive member of the opposite sex who always seems to be hanging around without ever actually having sex with anyone.
All this is in accordance with normal apathy calculus, due to the desirability of the goal. But, and here's the clever bit, the apathetic tunneling effect means that you can only be bothered to go straight back to bed. You are actually awake (sort of), mobile, and no longer in danger of exploding, and yet you still pass the ringing phone, the burning toast, and the rather promising-looking person making eyes at you as you pass. You can't stray from the apathy tunnel between bed and bog.