Maximum allowable temperature rise depends on the power dissipation and the heat loss so it is not a simple formula. There are two limitations on wire gauge- temperature rise and voltage drop. For instance, if your 10A circuit needs 16mm2 cable based on voltage drop, there will be absolutely no problem with temperature rise, as 16mm cable is rated 100A. It's worth checking that any calculated cable is OK for temperature rise, but it's usually a no-brainer.
On low voltage systems (12v), voltage drop tends to bite first. Generally with high voltage systems (240v), we work out cable size on temperature rise, and then do a voltage drop check as an afterthought. Personally, I prefer to work out resistance, then work out absolute drop, then work out percentage drop, but each to his own. The resistance of a length of pure annealed copper wire of cross section \$A mm^2\$ and length \$Lm\$ at room temperature is \$R=0.017\frac\$, but the first form keeps the correspondence between the factors and the way we derived it. I much prefer to build up understanding a step at a time.
What if you transcribe it incorrectly? Where do all the terms come from? Do we need all those factors of 100?
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