It depends on the length of the wire you're heating... and the voltage and current capabilities of your power supply.
I really don't know what your planned hot-wire cutter is going to look like or what you're planning to cut with it (tapered wings, 2d shapes, etc) and there's quite a bit more to this stuff than simply asking what size wire to use and whether it will melt or not. I'm old and tired and not prepared to go too far down this rabbit hole... so, I'll simply describe what I've done and you can adapt/ignore from there. This
web page covers this wire power stuff way better than I ever could so please refer to it. Hopefully it'll make sense to you.
I needed to cut 2" construction foam for a SIL project. A ~4" length of 30 AWG nichrome wire (I already had it on hand) will be plenty to handle that thickness.
View attachment 171920
For powering the wire at up to 1.5-2 amps (from the chart, 800*F to 1200*F), I needed at least 5 volts across ~2.5 ohms (resistance of 4.5" of 30 AWG nichrome wire). That's pretty readily available with a "power brick"-type supply... I had 12Vdc @ 6amps in my stash. If I used a fixed 12 volts, however, I'd need to power nearly 12" of wire (12v/6.7ohms = 1.8 amps) to stay in the 1.5 - 2 amp range. But I didn't want to power that long a wire... so used my Charsoon AntiMatter LIPO battery charger (from my RC days), with its "foam cut" (constant current) mode, to vary the voltage to set/maintain 2 amps (I believe other LIPO chargers have a "foam cut" mode as well) through the wire.
View attachment 171921
Here, the wire is powered... and the power supply has adjusted to 6.33 volts to provide the 2 amps through the wire. [Remember the connections to the nichrome wire have to be long enough to allow the wire to be positioned over the entire range of CNC motion so there's added resistance in the circuit connections to be made up for...]
View attachment 171922
And the wire is just faintly glowing red at 2 amps... and way more than required to cut foam...
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I started at 1.5 amps for the 2" styrofoam Tux I had cut out previously... but found that the construction foam (Foamular from HD) is more dense and upping the current to 2 amps allowed me to power through its 2" thickness and 800 - 1000 mm/min feedrate.
I genuinely hope this is helpful.
Good luck. -- David