How this instrument works
Ground speed is the aircraft's true rate of travel over the earth's surface — not the airspeed indicator's reading, which only measures motion through the surrounding air mass. The two agree only when there is no wind at all. Add moving air to a moving aircraft and the result is a vector sum: the true-airspeed vector and the wind's velocity vector, added tip to tail. The law of cosines closes that triangle algebraically without ever having to draw it, which is exactly what this formula does.
The angle θ is what turns the calculation into more than simple addition. Wind blowing exactly along the heading, θ = 0°, adds its full magnitude to ground speed. Wind blowing exactly against it, θ = 180°, subtracts its full magnitude instead. At any angle between those two, only part of the wind's speed acts along the flight path, and the cosine term measures precisely how much — the same reason a sail loses push the further the wind swings around toward the beam.
The result is a magnitude only, not a track. A crosswind component also pushes the aircraft sideways off the intended course, and a pilot corrects for that drift with a separate wind-correction angle applied to the heading — which changes θ itself. Ground speed and the corrected track are therefore normally solved together as one triangle, not read off this figure in isolation; treat this instrument as the speed half of that pair, not the whole of it.
- Enter True airspeed — the aircraft's speed through the air mass, from the airspeed indicator or a planned cruise figure, in knots, mph, or km/h.
- Enter Wind speed — the forecast or reported wind velocity at cruise altitude, in whichever unit the source gives it; the instrument converts internally.
- Set the Angle between heading and wind — how many degrees, or radians, the wind's direction of travel sits from the aircraft's heading.
- Read Ground speed — the actual rate of travel over the ground, which sets time en route and fuel burn for the leg.
Worked example — 120 kn TAS, a 20 kn wind at 45°
Take a true airspeed of 120 knots, a wind of 20 knots, and an angle of 45° between heading and the wind's direction of travel — the defaults this instrument opens with. Converted to the metres-per-second the formula runs in, that is TAS = 61.7333 m/s and W = 10.2889 m/s, with θ = 0.7854 rad. Plugging into GS = √(TAS² + W² − 2·TAS·W·cos(180° − θ)) gives GS = 69.3911 m/s.
Switch the result to knots and that reads 134.886 kn — noticeably above the 120 kn TAS, because at 45° the wind still keeps a sizeable component pushing along the direction of flight rather than against it. Compare the extremes with the same TAS and wind: a wind straight along the heading, θ = 0°, gives the full 140 kn of TAS + W, and one straight against it, θ = 180°, cuts ground speed to 100 kn, the TAS − W case.
Questions
Why isn't ground speed just true airspeed plus or minus wind speed?
Because wind rarely blows exactly along the heading. Simple addition or subtraction only holds at the two extremes, θ = 0° and θ = 180°; at any other angle, only the wind's along-track component changes ground speed, while the crosswind component pushes the aircraft sideways instead. The cosine term in GS = √(TAS² + W² − 2·TAS·W·cos(180°−θ)) accounts for that partial contribution automatically.
What exactly does the angle field measure?
The angle between the aircraft's heading — the direction the nose points — and the wind's direction of travel, not the compass direction the wind is reported as blowing from. Enter 0° for a wind blowing straight along the heading, 180° for one blowing straight against it, and 90° for a pure crosswind with no along-track component at all.
Does ground speed here include the drift a crosswind causes?
No — this formula returns a magnitude, how fast the aircraft covers ground, not the sideways drift a crosswind causes off the intended course. Correcting the heading to hold a course in a crosswind changes the angle between heading and wind, and with it the ground speed itself, so track-keeping and this speed figure are normally solved together, not one after the other.
How much difference can wind angle make to ground speed?
With a 120 kn TAS and a 20 kn wind, ground speed ranges from 140 kn with the wind directly behind, θ = 0°, down to 100 kn with it directly ahead, θ = 180° — a 40-knot spread from the same wind, just at different angles. At 45° it lands at 134.886 kn, closer to the upper end because that angle still keeps most of the wind's push along the flight path.
Why does flight planning need ground speed instead of just true airspeed?
Because time en route, fuel burn, and arrival time all depend on distance covered over the ground, and only ground speed measures that. An aircraft can hold a steady 120 kn TAS all day and still arrive noticeably early or noticeably late depending on the wind it meets, which is exactly why every pre-flight plan works this triangle leg by leg before departure.