Airfoil and wing geometry and nomenclature

Why it matters

Every aerodynamic number you will compute later — lift slope, zero-lift angle, induced drag, pitching moment — is tied to a geometric parameter of the section or the wing. If you cannot read a NACA designation, find the mean aerodynamic chord or convert a leading-edge sweep into a quarter-chord sweep, the theory that follows has nothing to stand on. Designers trade these parameters against weight, fuel volume and stall behaviour on every aircraft.

Key ideas

Airfoil (two-dimensional section)

• Leading edge (LE) and trailing edge (TE): the front-most and rear-most points of the section.
• Chord line: the straight line joining LE and TE; its length is the chord c. All section coordinates are usually written as fractions of c (x/c, y/c).
• Mean camber line: the locus of points midway between the upper and lower surfaces, measured normal to the camber line itself. It starts and ends on the chord line at LE and TE.
• Camber: the maximum distance between the mean camber line and the chord line, written as a percentage of chord, together with its chordwise location. A symmetric airfoil has zero camber; its camber line is the chord line.
• Thickness: distance between upper and lower surfaces measured perpendicular to the camber line; the maximum value t is quoted as t/c (thickness-to-chord ratio).
• Leading-edge radius and trailing-edge angle complete the description. A blunt LE delays leading-edge separation; a sharp TE is what makes the Kutta condition (later topic) applicable.
• Angle of attack α: angle between the chord line and the free-stream velocity. The absolute angle of attack is measured from the zero-lift line instead, so α_abs = α − α_L=0. For a positively cambered section α_L=0 is negative.

NACA designations

• 4-digit NACA MPXX: maximum camber M % of chord, located at P/10 of chord from the LE, maximum thickness XX % of chord. NACA 2412: camber 2 % c at 0.4c, thickness 12 % c. NACA 0012 is symmetric, 12 % thick.
• 5-digit and 6-series designations encode design lift coefficient and (for the 6-series) the extent of favourable pressure gradient for laminar flow; the exact meaning of each digit should be taken from the series definition in your data book.

Wing (three-dimensional) geometry

• Span b: tip-to-tip distance, measured perpendicular to the plane of symmetry.
• Planform (reference) area S: projected area of the wing on the plane containing the chord, normally including the part that passes through the fuselage.
• Root chord c_r and tip chord c_t; taper ratio λ = c_t / c_r (λ = 1 rectangular, λ = 0 pointed delta).
• Aspect ratio AR = b² / S; for a rectangular wing this reduces to b/c. High AR means low induced drag (gliders, AR 20–30); low AR means a stiff, light, compact wing (fighters, AR 2–4).
• Mean geometric chord c̄ = S / b and mean aerodynamic chord (MAC): the chord of an equivalent rectangular wing having the same lift and pitching moment. Moments and centre-of-gravity positions are quoted as % MAC.
• Sweep angle Λ: angle between a constant-chord-fraction line (LE, quarter chord, TE) and the line perpendicular to the plane of symmetry. Always state which line: Λ_LE, Λ_c/4 and Λ_TE differ unless λ = 1.
• Dihedral Γ: upward angle of the wing from the horizontal, seen from the front; it gives lateral (roll) stability. Negative dihedral is anhedral.
• Twist: geometric twist is a change in section incidence along the span; aerodynamic twist is a change in zero-lift angle (different sections). Washout (tip at lower incidence than root) makes the root stall first and keeps the ailerons effective.

These parameters connect directly to later topics: AR and λ set the spanwise lift distribution and induced drag (lifting-line theory), Λ sets the effective normal Mach number and the spanwise flow (swept wings), and camber sets α_L=0 and C_m,ac (thin airfoil theory).

Formulas

• AR = b² / S — b span (m), S planform area (m²); dimensionless.
• λ = c_t / c_r — c_t tip chord (m), c_r root chord (m); dimensionless.
• S = (b/2)·(c_r + c_t) = (b·c_r/2)·(1 + λ) — straight-tapered (trapezoidal) wing.
• c̄ = S / b — mean geometric chord (m).
• c_MAC = (2/3)·c_r·(1 + λ + λ²)/(1 + λ) — mean aerodynamic chord of a trapezoidal wing (m).
• ȳ_MAC = (b/6)·(1 + 2λ)/(1 + λ) — spanwise distance of the MAC from the centre line (m).
• tan Λ_n = tan Λ_m − (4/AR)·[(n − m)·(1 − λ)/(1 + λ)] — converts sweep of the m-chord line to the n-chord line (m, n chord fractions, e.g. 0 for LE, 0.25 for quarter chord, 1 for TE); trapezoidal wing only.
• α_abs = α − α_L=0 — angles in degrees or radians, consistently.

Worked examples

Example 1 (standard). A trapezoidal wing has span b = 20 m, root chord c_r = 5 m and tip chord c_t = 2 m. Find S, AR, λ, c̄, c_MAC and its spanwise position.

1. S = (b/2)(c_r + c_t) = (20/2)(5 + 2) = 70 m².
2. AR = b²/S = 400/70 = 5.71.
3. λ = c_t/c_r = 2/5 = 0.4.
4. c̄ = S/b = 70/20 = 3.5 m.
5. c_MAC = (2/3)(5)(1 + 0.4 + 0.16)/(1.4) = (3.333)(1.56/1.4) = 3.71 m.
6. ȳ_MAC = (20/6)(1 + 0.8)/(1.4) = 3.333 × 1.286 = 4.29 m from the centre line.

Answer: S = 70 m², AR = 5.71, λ = 0.4, c̄ = 3.5 m, c_MAC = 3.71 m at 4.29 m from the centre line. Note c_MAC > c̄ for a tapered wing.

Example 2 (GATE level). A trapezoidal wing has AR = 6, λ = 0.5, b = 12 m and leading-edge sweep Λ_LE = 30°. Find the root chord, tip chord, quarter-chord sweep and trailing-edge sweep.

1. S = b²/AR = 144/6 = 24 m².
2. From S = (b·c_r/2)(1 + λ): c_r = 2S/[b(1 + λ)] = 48/(12 × 1.5) = 2.67 m; c_t = λ·c_r = 1.33 m.
3. Quarter chord (m = 0, n = 0.25): tan Λ_c/4 = tan 30° − (4/6)(0.25 × 0.5/1.5) = 0.5774 − 0.0556 = 0.5218, so Λ_c/4 = 27.6°.
4. Trailing edge (n = 1): tan Λ_TE = 0.5774 − (4/6)(1 × 0.5/1.5) = 0.5774 − 0.2222 = 0.3551, so Λ_TE = 19.6°.

Answer: c_r = 2.67 m, c_t = 1.33 m, Λ_c/4 ≈ 27.6°, Λ_TE ≈ 19.6°. Sweep decreases from LE to TE because the wing tapers.

Common mistakes

• Using AR = b/c for a tapered wing. It is b²/S; b/c holds only for a rectangular wing.
• Quoting "the sweep" without saying which line. GATE problems often give Λ_LE and need Λ_c/4.
• Measuring camber or thickness as a fraction of span instead of chord.
• Reading NACA 2412 as "camber at 4 % chord". The second digit is in tenths of chord: 0.4c.
• Taking taper ratio as root/tip; by convention it is tip/root and lies between 0 and 1.
• Confusing mean geometric chord S/b with the mean aerodynamic chord; they are equal only for λ = 1.
• Thinking a symmetric airfoil cannot lift — it lifts at any non-zero α; it just has α_L=0 = 0.

For GATE AE

Expect direct numericals on AR, taper ratio, area and MAC of a trapezoidal or delta planform, conversion between LE and quarter-chord sweep, and decoding NACA 4-digit designations into camber, its location and thickness. Conceptual MCQs test definitions (mean camber line, absolute angle of attack, washout, dihedral). Practise sketching a planform from the given numbers before calculating — most errors come from a wrong picture.

Quick check

1. Decode NACA 4415.
2. A rectangular wing has span 10 m and chord 1.25 m. What is AR?
3. A delta wing (λ = 0) has root chord 6 m and span 8 m. Find S and AR.
4. For a cambered airfoil with α_L=0 = −2°, what is the absolute angle of attack at α = 4°?
5. What is washout and why is it used?

Answers: 1. Max camber 4 % c at 0.4c, thickness 15 % c. 2. AR = 8. 3. S = 24 m², AR = 64/24 = 2.67. 4. 6°. 5. Reduced incidence towards the tip, so the root stalls first and the ailerons stay effective.