focal length, aperture diameter & f/stop

something is confusing to me about the whole f/stop calculation: f/number = focal length / aperture diameter.

I have a Canon EF-S 55-250 lens, at 55mm focal length it has a max f/stop of 4.0. Using the f/number equation, I can determine that the maximum aperture diameter is 13.75mm (55/4)

At 250mm focal length, the lens as a max f/stop of 5.6. Again, using the f/number equation the aperture diameter at this focal length is 44.64mm (250/5.6). This is over 3 times wider than the diameter at 50mm focal length!

How can this be? Why can't the lens keep the 44.64mm diameter at 55mm and give f/1.2?

Clearly this isn't the case as there are no lenses with these types of specs. But I feel the explanations of f/stop from various resources (books or interweb) only talk about f/stop at a given focal length with a varied aperture diameter.

I have yet to see an explanation of how focal length changes f/stop for a given aperture diameter. My guess is it's not that simple.

Les Olson wrote:

The f number is not, in fact, the ratio of focal length to the diameter of the physical diaphragm, although it is often presented like that for simplicity's sake. It is the ratio of the focal length to the "entrance pupil", which is the image of the diaphragm when you look through the front of the lens. If there are no optical elements in front of the diaphragm the entrance pupil and the physical diameter are the same, but in a camera lens what you see is a virtual image, which may be magnified or diminished relative to the real diaphragm.

Les is right.
Taken from wikipedia:

The entrance pupil is the image of the physical aperture, as seen through the front of the lens. The size and location may differ from those of the physical aperture, due to magnification by the lens.
http://en.wikipedia.org/wiki/Entrance_pupil

Now take out your lens and a ruler. Take the lens off the camera set it at most tele and measure the apearant size of the light part of the lens (where you see the light coming through from behind). Note this size in mm. Then turn on your zoom-ring and see what happens with this light part! Measure and note the size again on the widest setting of the zoom ring.

The reason for the changing of the size is that some lenses inside the lens and the actual opening are moving relativly to each other.

Now do the arithmatic. It will work (wel, it's hard to measure exact enough, but you'll come close).

You can do this looking, measuring and arithmatic with other lenses too. On a zoom lens with fixed maximum aperture the entrance pupil will change in size too.