I'm making a corner shelf for spare toilet paper from a thin plywood sheet, say 5mm.
I want to connect the plywood to the corner walls with a bolt/nut joint like this:
My question is -- how do I calculate the size of a cutout for such a joint so that a wrench would fit?
What you're looking for is called "Wrench Clearance". There are tables for these in Machinery's Handbook (pp 1344-1355 in 24th ed). There are also resources online like this table for box end wrenches or this one for open end wrenches
I will caution you that the clearances listed in these types of tables should be considered minimums and more is generally better within reason of course.
I also think that you'll find this joinery marginal for plywood this thin. It's much better suited for 1/2" - 3/4" plywood.
The external size of wrenches can vary quite a lot. Ring spanners tend to be the slimmest but you need enough room to get them off the end of the bolt once it is tight. You can also get ring spanners with slots which fit over a bolt shaft but theses are a bit more chunky.
So really you just need to measure the size of the wrench you want to use.
Having said that you might want to consider using a captive nut, this will make assembly easier and you may well find that it is more stable and distributes the load better for this sort of installation in wood. It doesn't take much for a hex nut to dig into plywood.
Indeed you may find that 5mm plywood is really a bit thin to use this sort of construction method at all.
