Wind is a tax, never a wash
The intuition that a headwind out is repaid by a tailwind back is wrong, and the reason is that you spend longer in the slow leg than in the fast one. Work the algebra and the out-and-back radius comes out as the still-air radius multiplied by one minus the square of the wind fraction.
With the defaults — 15 m/s airspeed, 5 m/s wind, 12 minutes of flying after a 2 minute reserve — still air would give 5,400 m and the wind gives 4,800. The wind is a third of the airspeed and it costs a ninth of the radius, which is 11.1 percent. That squaring is the whole shape of the problem: gentle at first and then very fast.
The turn point is nowhere near the halfway time
Go downwind first at 20 m/s and after six minutes — half the flying time — you are 7,200 m out. Getting back into 10 m/s of ground speed takes 12 minutes, and you had 6. The correct turn is at 4,800 m, reached in 4 minutes, leaving 8 minutes for the way home. A third of the time out, two thirds back.
That asymmetry is the part worth internalising, because nothing on a screen shows it. Ground speed reads 20 on the way out and everything feels fast and cheap. The clock and the map disagree by a factor of two, and the disagreement is invisible until you turn round.
Across the wind is the middle case
A track at right angles to the wind is flown crabbed, and the ground speed along the track is the airspeed with the wind component taken out in the Pythagorean sense — 14.14 m/s for the defaults. That gives a 5,091 m radius, between the 4,800 along the wind and the 5,400 of still air.
Practically, that makes the survey line orientation a real decision on a windy day rather than an aesthetic one. Lines flown across the wind cost less than lines flown along it, though the turns at each end are harder and the aircraft holds a constant crab angle the whole way down.
What the geometry does not include
Power. Holding 15 m/s of airspeed into a headwind takes more current than holding it in still air, because the aircraft sits at a larger tilt angle and is working against more drag, so the endurance that feeds this page is itself lower on a windy day. Gusts, wind gradient with height, and the fact that wind at 100 m is usually stronger than the wind you measured standing on the ground are all outside this too. So is anything about what you are permitted to do — how far out, in what airspace, in sight of whom — which is set by your own aviation authority and not by arithmetic.
Questions people ask
Does a tailwind out cancel a headwind back?
No, and it never can. You spend longer in the slow leg than in the fast one, so the round trip always loses. The radius is the still-air radius times one minus the square of the wind-to-airspeed fraction: wind at a third of your airspeed costs 11 percent, at half it costs 25 percent, at three quarters it costs 56 percent.
Where is the turn point when flying downwind first?
Much closer than half the time. At 15 m/s airspeed in 5 m/s of wind with 12 minutes to spend, the turn is at 4,800 m after 4 minutes, leaving 8 minutes for the way home. Turning at the six-minute mark puts you 7,200 m out needing 12 minutes to return with 6 available. The clock and the map disagree by two to one and nothing on the screen shows it.
Is it better to fly survey lines along the wind or across it?
Across, on the arithmetic. A crosswind track holds a ground speed of the airspeed with the wind taken out in the Pythagorean sense — 14.14 m/s instead of 15 — which gives 5,091 m of radius against 4,800 m along the wind. The cost is a permanent crab angle and harder turns at the ends of each line.
What happens when the wind is stronger than the aircraft?
There is no upwind leg at all. Ground speed into the wind is zero or negative, so no distance downwind is recoverable at any endurance, and the page says so rather than printing a number. Crosswind tracks stop existing at the same point, because the crab correction needs the airspeed to exceed the wind.
Does this include the extra current used flying into wind?
No — this page is pure geometry, and it takes your endurance as given. Holding an airspeed into a headwind also costs more power than holding it in still air, which shortens the endurance that feeds this calculation. Work the endurance out for a windy day first, put that lower figure in here, and the two penalties are counted separately rather than one being forgotten.