Haven't a clue meself but did find in days of old that if you were in a particularly new estate the local carpet suppliers could tell you off the top of their head exactly how much was required. This might even apply to older houses and a quick call might provide the info.
If it's a standard staircase - ie no return on the stairs just measure the hall/landing. We bought carpet last year and that's what we did. Some shops - like Arnotts insist on measuring themselves beforehand although this is unusual.
Thing is, if you get the supplier to measure themselves, there'll be less waste, or even less/no chance of not having enough carpet. If they do it for free why not get them to do it?
If it's a standard staircase - ie no return on the stairs just measure the hall/landing.
My dad does this and it is definitely not something I would recommend you do yourself (unless it is very very straightforward). You should see the jiggery pokery he does to get the best possible usage from the carpet, ensuring that pile runs in the same direction (to avoid colour variance) and all kinds of palaver to obtain the most professional finish. I grew up watching him do it and still wouldn't feel confident doing the complicated ones without consulting him.
We measured the hall, landings (three storey house) and returns and counted the number of steps...this was all the carpet shop required and there wasn't a lot of waste.
I bought carpet recently to fit out stairs / landing / bedrooms - the carpet shop came and did all the measuring. There was very little waste and it was part of the service that they did that.
A standard staircase has 13 steps, most carpet shops base their prices on this. So all you really need to do is measure the hall and landings. But, as others have said, if they offer a free measuring service let them do the work for you.