Intuition The one core ideaA real wing has ends , and at those ends air spills from the high-pressure bottom to the low-pressure top, curling into whirlpools that trail behind and push air downward. This downward push tilts the lift backward into a drag — and the longer and thinner the wing, the smaller the fraction of it that suffers this leak, so the drag shrinks.
This page builds every letter, ratio, and picture the parent note Aspect ratio — effect on induced drag leans on — starting from a smart 12-year-old who has seen none of it. Read top to bottom; each block only uses symbols defined above it.
Before any formula we need three simple lengths and one area. Picture the wing as a flat board seen from directly overhead (bird's-eye view). This flat outline is called the planform .
b span b Picture: the horizontal red arrow stretching across the whole wing in the figure.
Why we need it: leakage happens only at the two tips . A wing's "how much tip do I have per unit of wing" question starts with knowing how wide it is.
c chord c Picture: the short black arrow pointing from nose-edge to tail-edge in the figure.
Why we need it: the chord tells us how "deep" (front-to-back) the wing is. A fat stubby wing has a big chord; a skinny wing has a small one.
S planform area S Picture: the shaded region inside the outline.
Why we need it: lift is spread over this whole area, so every force we compute later gets shared across S
Now we can build the star of the topic. We want a single number that says "is this wing long-and-thin or short-and-fat?" — independent of how big the wing is overall.
ratio and not just the span
A big airliner and a tiny model glider can both be "long and thin." What matters is not the raw span but span compared to chord. A ratio cancels out overall size and leaves only the shape .
A R A R = S b 2 S = b c A R = b c b 2 = c b A R how many chord-lengths fit along the span .
Picture: in the figure, the skinny red wing fits many chords across its span (high A R A R
Recall Why square the span instead of usingb / c b 2 / S any planform shape, while b / c c b c A R
Worked example Reading real numbers
A sailplane: A R ≈ 30
A fighter jet: A R ≈ 3
The whole topic hinges on a force that gets tilted . So we must first pin down what lift is and what "tilt an arrow" means.
Definition Freestream velocityV V Picture: the long horizontal black arrows approaching the wing from the left.
Why we need it: drag is always measured along this direction — it is the "backwards" that a plane fights against.
L Lift is the force pushing the wing upward , perpendicular to the oncoming air. It comes from lower air pressure on top of the wing and higher pressure below.
Picture: the vertical arrow. In the figure, the black lift arrow points straight up because the flow is horizontal.
Why we need it: lift is the only reason induced drag exists — no lift, no leak, no drag.
D D i Drag is any force opposing motion — pointing backward along V Induced drag D i specific drag that appears only because a finite wing makes lift .
Picture: the small red horizontal arrow pointing backward — it is the shadow (projection) of a tilted lift arrow onto the flight direction.
Intuition Perpendicular towhat , exactly?
Here is the subtle heart of the topic. Lift is always perpendicular to the local airflow the wing actually feels. If the tips force the local air to tilt downward, then "perpendicular to the local flow" is no longer straight up — the lift arrow tips backward, and that backward lean is the induced drag. Section 5 makes this precise.
Now the physical machinery that does the tilting.
Definition Trailing vortexA trailing vortex is a tube of swirling air that peels off each wingtip. It forms because high-pressure air under the wing sneaks around the tip to the low-pressure top, and that sideways-then-upward escape rolls the flow into a spiral.
Picture: the two red spirals streaming behind the tips in the figure.
Why we need it: these spirals are the engine of induced drag — see Trailing vortices & downwash .
w Downwash w downward air velocity created behind the wing by those two spinning vortices.
Picture: the short downward arrows in the middle region between the vortices.
Why we need it: w
Common mistake "The vortex is just harmless swirling."Why it feels right: swirls look like decoration behind a wing.
The fix: spinning that air costs energy every second. That continuous energy cost, felt by the wing, is the induced drag. It exists even in a perfectly frictionless fluid — it is inviscid .
The tilt is an angle , so we need one trigonometric tool. We pick exactly the one that answers our question — no more.
tangent enters here
We know two velocities: the forward speed V w right triangle . The angle of tilt sits between the true forward direction and the slightly-drooping local flow. The ratio that links these two known sides to that angle is exactly the tangent — "opposite over adjacent" — which is why tan
α i α i alpha , subscript i downward relative to the freestream.
tan α i = V w ⟹ α i = tan − 1 ( V w ) tan − 1 "which angle has this tangent?" It undoes the tangent to hand us the angle itself.
Picture: the red angle wedged between the black horizontal V
Recall Which cases could break the small-angle trick?Very slow flight (landing, high lift) makes w α i w = 0 α i = 0
The parent suddenly writes C L C D , i L D i
Definition Dynamic pressureq q = 2 1 ρ V 2 ρ rho ) is the air density (mass of air per cubic metre). q Why we need it: dividing a force by q S
Definition Lift and induced-drag coefficientsC L = q S L , C D , i = q S D i C L C D , i Why we need them: they let us compare a paper glider and a jumbo jet on the same footing — geometry (A R
Mnemonic Coefficient = force ÷ (push × area)C = q S force q S shape's opinion.
Definition Span efficiencye e 0 1 ideal (elliptical) one. Ideal ⇒ e = 1 e ≈ 0.7 0.9 Why we need it: the clean formula C D , i = C L 2 / ( π A R ) best possible case. Real wings do slightly worse, and e Elliptical lift distribution .
Aspect ratio AR = b squared over S
CDi = CL squared over pi e AR
Every arrow here is a symbol you can now read. Together they assemble the parent's headline result C D , i = C L 2 / ( π e A R )
Test yourself — reveal only after you have an answer.
What does the span b The tip-to-tip width of the wing
What is the planform area S The area of the wing's overhead (bird's-eye) outline
Write aspect ratio for any wing, then for a rectangular wing A R = b 2 / S A R = b / c
In plain words, what does A R How many chord-lengths fit along the span (long-thin vs short-fat)
What direction is lift, and relative to what? Perpendicular to the local airflow the wing actually feels
What creates the trailing vortices? High-pressure air below leaking around the tips to the low-pressure top
What is downwash w The downward air velocity behind the wing caused by the tip vortices
Why does tan V w = w / V = tan α i
State the small-angle shortcut For tiny α i α i ≈ tan α i ≈ sin α i
Why divide forces by q S To remove air density, speed and size, leaving a pure shape number
What is q Dynamic pressure q = 2 1 ρ V 2
What does span efficiency e e = 1
