How many dimensions does space have?

I’ve been a CADist for many years now and have recently been teaching myself 3d CAD, anyway, to calculate volume you need 3 dimensions X,Y,Z or length, width, height etc. This really only works for what you might call standard geometric solids, once you start adding curves (never mind compound curves) all hell brakes loose, and an approximate volume based on Pi is used.

So it occurred to me that to find the exact volume of a sphere, or portion thereof, you need to add more and more dimensions/lines of dissection to gain a more accurate volume, until you get to the point where you would need an infinite number of dissection lines to gain a perfectly accurate volume.

So the question is, why do we say space is 3 dimensional, and not infinitely dimensional?