What are you sticking it to?
(Our awning “floor” was one end of a huge ARCO tarpaulin that went under the whole tent, it got pegged into the grass via the eyelets.)
'Groundsheets' and 'Carpets' tend to be very different things!
Typical groundsheets (tarpaulin type material) are made from polyethylene, which is the devils own job to stick, as the material is inherently resistant to surface treatments like adhesives and paints etc. Commercially they would normally be heat welded to join and/or stitched, or treated (surface modifiers) in ways not available to the DIY types for enduring marking with inks/paints.
As a DIY solution it tends not to get any better than quality Gaffer or double sided tape used for waterproof/gas proof floor membrane sheeting.
Proper awning Carpets come in a number of types, so would need to know the specifics to advise on those. That said, many are polypropylene based too, so adhesives may not be suitable.
If it is a tarp, then no need to cut, just tape down with Gorilla tape on the underside to the required shape.
Need more information about the material and construction of the carpet to be able to advise.
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Your only real choice is sewing if it's the kind of coarse weave 'carpet' I think it is.
Almost certainly too heavy a job for most domestic sewing machines (probably wouldn't get multilayers under the foot!), and you'd need a 'walking foot attachment' as the fabric feed would struggle on it's own. A sailmaker, tent repairer or the like would likely be able to do it for a nominal sum.
An alternative may be Eyelets placed a lot closer together than would be the norm for simple anchor points, say every 300-400mm. Do a 3 layer hem so no raw material edges exposed. Plastic 'no setting tool' (need a hammer though) eyelets, and metal eyelet kits with setting tool are fairly easy to find.
I use my carpet in two different sized awnings, in the smaller one I simply fold the excess under to make it two layers thick.
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