Yes, that is the correct wire gauge. However, if your tub is much more than 100 ft. from the panel you might consider going up in wire size to #4 due to voltage drop.
# 10 bare copper.
Amperage is always measured with the guage in series with the hot wire. That is the one wire is cut and the guage goes in line as opposed to some devices which have a hot and neutral (or ground).
THE HOT TUB IS TO RELAX
You use a hot tub cover after the hot tub has cooled off. Using it while your hot tub is still warm can cause your cover to break.
A 12 gauge wire can support a load of approximately 2200 watts.
Aluminium wire has produced problems due to unequal expansion/contraction at the brass or copper terminals and long-term surface oxidation. For that reason copper is recommended and there is usually no good reason not to use copper (of the correct gauge for the current).
#6 3 conducter
The size of the wire is stated by its gauge under American Wire Gauge. Six gauge wire is size 6 AWG.
Wire gauges are defined in such a way that the lower the gauge, the thicker the wire. So, 8 gauge wire is thicker than 10 gauge wire.
What about "pool chemicals in a hot tub"?
for a luxury hot tub it is 15,000