Yes, 6 AWG copper cable will give you a 20A circuit, at 60 cents a foot x 3 conductors x length.
A better choice is 4 AWG Aluminum for 1/3 the price. Yes, aluminum is safe in these large wire sizes.
That's awfully heavy wire for a couple LEDs.
You're not legally required to upsize the cable. As long as the breaker is correct for the wire size (i.e. 20A breaker for 12 AWG wire), it will protect the wiring from overheating, and that is the purpose of breakers.
It's a big mistake on those "voltage drop calculators" to give the amperage of the breaker instead of the amperage of the practical, real-world load. The latter is the one you're concerned with. It's OK to temporarily load the circuit beyond that, so long as the breaker is correct for the wire. In fact, you already do that - how do you get power across the yard? With a long orange extension cord. Look at yours - is it hooty-tooty 6 AWG? Of course not, I bet it's 16 AWG, or 14 at most. And yeah it gets warm - I've melted plenty of snow with mine. The tools got a lot of voltage drop, didn't fry, and we all lived to tell the tale.
So if it's OK to run 16 AWG extension cords, why do you need 6 AWG if it's in conduit? YOU DON'T. You do need to protect the wire though, which is why you need 12AWG wire if the breaker is 20A.
Consider an extreme scenario: a 1000-foot run of 12 AWG to run a
couple of post lights, "and a convenience outlet". 2x 15W LED post
lights. 30 watts total, at 120V that's 1/4 amp or 0.25A. Now, go and
punch that into the voltage drop calculator and allow a 99% drop
just to see what happens:
Huh what? 1% drop? The calculator goes to minimums - since it's
for Code electrical, minimum wire size is 14 AWG. This gives a 1%
drop. 14AWG is more than adequate even for a 1000 foot run.
So what happens if our kid is going to school, we put a shelter there,
and put a 1500W heater in the convenience outlet? That actually requires some
sharp-pencil work with Ohm's Law, but it boils down to the circuit flowing 8.03A, the heater
giving 618W of useful heat while 345W is lost in the wiring (0.345
watts per foot, no big deal). And remember that's with 14AWG wire
going 1000 feet, which is pretty extreme.
Just for chuckles, go back and plug 20A and 3% voltage drop into the calculator. It'll spec 3/0 copper or 300MCM aluminum. Now price 3000' of that.
I'm only suggesting 12 AWG wire going 120 feet.
So my best answer is, don't worry about it. Breaker it for 20A, or 15A if it really bothers you.
Best of both worlds: reduce voltage drop further with 240V.
OK then, shift gears. Instead of 12AWG, run 14 AWG. what!?
Run 2-wire (hot-hot) 240V power. Fit a NEMA 6 receptacle.
You're skeptical. I get it.
Now add a step-down transformer. This plugs into the 240V source and gives 120V output. Half the voltage, double the amps, or vice versa. If we plug in a huge 16A 120V tool, it's only going to draw 8A at 240V. Go back and plug that into the voltage drop calculator.
So what do you do about the lights?
- many LED fixtures and most new fluorescents will accept 240V. (you're not allowed to have E26 Edison bases on 240V lighting, however.)
- Run a second 120V circuit - 14 AWG is ample, as you saw from the 1000 foot example above. The two circuits can share the ground wire (4 wires + ground)
- Run hot+hot+neutral (3 wires + ground) put in a sub-panel. Feed the lights with 120V and the transformer with 240V. Yes, on 15A fed with 14AWG. And then you can brag that you did that!
Now you have a wire size, choose conduit
If we were talking about wrestling big old 4 AWG aluminum, I woulda said a 2" conduit if not 3", just to make it possible to pull without hiring an electrician for his truck full of varsity pulling tools. But now that we're talking 14AWG, that's so easy to work with, just get it over there any practical way... bury a little (if you bury, do use large conduit for future expansion)... staple it to the fence, whatever's legal... and if it breaks, splice it.
It really boils down to money. What's the least expensive way that'll do what you want legally, and give you future headroom?