That's how I think of it too Pops, the lens of a given focal length makes a certain size image circle which the sensor, either FF or APS-C sits inside.
Originally Posted by PopsPhotos
Since the APS-C sensor is smaller, some of that gets wasted around the edges, the sensor only registers a (virtual) crop from the whole image circle, thus it appears as if you have used a longer focal length lens (compared to a FF/film camera).
To slightly complicate matters, a DX lens 'doesn't bother' to make the large image circle, but crucially, the focal length isn't different, so you still apply the same thinking to it.
That also neatly allows you to figure why vignetting isn't an issue with a FF lens on a DX body and why the performance is better - because you're only using the good bit in the middle. Someone with FF lenses, moving from DX body to FF body may notice this loss of quality (in addition to the change of angle of view).
It may help you visualise this if you prove it to yourself as follows;
a) Get a sheet of white paper
b) Draw two boxes on it; one 24mm x 36mm and one centrally inside that, 16mm x 24mm (use a pen to aid visibility) - these are your sensors
c) Point a bright desk lamp at a patterned wall covering or magazine cover propped against a wall - this is your subject
d) Using 50mm f/1.8 or similar (ideally prime) lens
e) Turn off other room lights, shield the paper from light spill
f) Move paper and lens to focus an image the wall covering (or magazine, etc.) onto the paper
g) If you can hold it still enough, observe how the scene captured by the inside box is 'more zoomed in' than the outside box
That's why it is called crop factor
Notes to improve effectiveness
Adapt experiment for what you have available
Make sure the lens is not stopped down; do what ever you have to to get its aperture wide open
Keep 'sensor' paper parallel to 'subject'
Get someone with steady hands to help, so you can concentrate on looking at the results.