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Wind Triangle Ground Speed

Also known as ground speed from true airspeed · wind triangle · headwind component · tailwind component · E6B ground speed · speed made good over the ground

Vg=V2−W2sin⁡2θ−Wcos⁡θV_g = \sqrt{V^{2} - W^{2}\sin^{2}\theta} - W\cos\theta

Worked example: 150 kn TAS, 30 kn wind 60° off the nose → 132.73 kn ground speed — press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Try different units for next level excitement!

See also
Wind Correction Angle — The other half of the same triangle — how far to crab to hold the track.Estimated Time En Route — This ground speed is the one that sets the fuel and the arrival time.Set and Drift of the Current — The marine version of the same vector problem, with a current instead of a wind.
Here the solver did the work — could you?

The wind triangle →

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Wind Triangle Ground Speed explained

VWVgθ

An aircraft flies in a parcel of air that is itself moving over the ground, so its path over the ground is the vector sum of two things: where it is pointing and how fast through the air, and where the air is going and how fast. Draw those two as arrows head to tail and the third side that closes the triangle is the track and ground speed. That is the wind triangle, it is the entire content of the E6B flight computer, and it is the same triangle a mariner draws for a current — the labels change and the geometry does not.

The angle θ\theta used here runs from the intended track to the direction the wind is blowing FROM, so 0° is a dead headwind and 180° a dead tailwind. Wind is always named for where it comes from — a westerly blows from the west toward the east — and this is a genuine source of sign errors, because a current is named the opposite way, for the direction it sets toward. If you plot a wind and a current on the same sheet, write down which convention each arrow is drawn in before you do anything else.

Only two pieces of the wind actually do anything. The crosswind component Wsin⁡θW\sin\theta has to be cancelled by pointing the nose into it, and that is the wind correction angle. What remains of the airspeed after that correction, V2−W2sin⁡2θ\sqrt{V^{2}-W^{2}\sin^{2}\theta}, then has the along-track component Wcos⁡θW\cos\theta added or subtracted. Notice what this means: a strong headwind costs you speed but needs no correction at all, while a moderate beam wind needs a large correction and costs comparatively little speed.

Two mistakes to name. The first is using indicated airspeed where true airspeed belongs — at altitude the two differ by well over ten percent, and a flight plan built on the indicated figure is optimistic in the worst possible way. The second is the round-trip intuition: pilots and sailors both tend to assume a wind that helps one leg and hinders the return comes out even. It never does. You spend longer in the headwind than in the tailwind, so the headwind gets more time to hurt you, and a round trip in any wind at all always takes longer than the same trip in still air.

Wind Triangle Ground Speed formula

Vg=V2−W2sin⁡2θ−Wcos⁡θV_g = \sqrt{V^{2} - W^{2}\sin^{2}\theta} - W\cos\theta
Where
  • VgV_g= Ground speed (kn)
  • VV= True airspeed (kn)
  • WW= Wind speed (kn)
  • θ\theta= Wind angle off the track (°)

Missing one of these? Work it out first, then come back

Part of this formula set

Planning a leg

6 formulasTwo waypoints to a heading and a time: great circle distance and bearing, the compass correction, the wind triangle, and the time en route.