Building a sub in a Ikea Besta

I have never build a sub before, but would like to build a hometheater sub in a Ikea Besta cabinet.
Would a 56x36x42 be enough for low hitting sub?
Or would I need two cabinets combined?
I was thinking of a downfiring sub.

Any hints or tips are welcome.
(For example which driver to take :))
 
I was thinking of making the box in the cabinet.
Not use the cabinet itself.
Yes dimensions are in cm ;)

It can be done, but the construction guidelines still apply. The driver is is irrelevant because the situation cannot be realistically modelled. To give you an example . . . In the bedroom I had an 8" sub in a cube cabinet. My partner complained that it was ugly, fortunately it was a snug fit into the bottom of the night-stand. Result: no sub-bass, but a horrific peak around 80Hz making everything sound like it was being played at an 80s wedding reception. Most calculations are made using the assumption that frequencies travel exclusively through air - they do not. Lower frequencies travel by all and any means.

The subwoofer argument is eternal: do you want 'true' bass or 'more' bass. Lock a subwoofer inside your freezer and your audible bass will vanish. But the items on your cupboard shelves tell you : your 15Hz response is phenomenal!

Too deep?

The answer is try it and see
 
Is this a floating cabinet? You will add a lot of weight to the construction. I would put a ported 12" subwoofer in there, as a sealed 15" might not fit given your measurements.

I would first try if this is the best location for your sub before building it.

And you might not need to build a cabinet in a cabinet, putting in a solid back and some 45's in the corner should get it airtight, add some dovel panel bracing and you are good.
 
Some interesting calculations: A 4'x8' sheet of Formica laminate is about the same price as a Besta cabinet, but the outside is only about 13 sq ft, less than half a 4x8 sheet. I would probably just make a sub that works and cover it, or some other finish.
Also, big speakers in a small box sound bad. Efficiency aside, decide how big the box can be and then pick a (smaller) driver that works in that size box. Cheap 2.1 computer speakers have amazing bass for their size. Note they are always ported.