Perhaps you could look at a picture of a bassoon, preferably with the parts labelled, and then describe to us which dimension you want. Do you mean the diameter of the tube (which of course varies continuously along its length)? If so, do you want the internal or the external diameter?
The reason we are all having trouble with your request is that we are not sure which part you need measured. If you have seen a bassoon, you will know that it consists of a bottom piece, and two other pieces come out of it, much like a tree which has a short trunk with two upright branches. Furthermore, the "branches" are not the same width from top to bottom, and neither is the "trunk." If you check out wikipedia.org and look up bassoon you will find more information.
OK - diameter: If you've looked at a picture (or better still a real bassoon) you'll know that it starts off very thin at one end and gets wider all along its length. The narrow end of a crook (the curved metal tube that you fix the reed on to blow into) is 4mm diameter on mine. The internal diameter at the other end is 42mm. There's a plastic ring around the opening at that end that makes the overall external diameter quite a bit more, and below that the external diameter is variable because it's got a kind of swelling in it. Confused? You need to see one!