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Aerospace
Trapezoidal wing planform area
Planform area of a symmetric, linearly tapered wing of full span b with root chord cr and tip chord ct: the average chord times the span.
- Ref
- AER-18
- Rev
- v1
- Field
- Aerospace
- Result
- S [m²]
A linearly tapered wing is a trapezoid, so its area is the span times the average of the root and tip chords: S = b(cr + ct)/2. The planform area is the projected area, and the root chord is measured at the centreline. S is the reference area for lift and drag coefficients.
Planform area of a symmetric, linearly tapered wing of full span b with root chord cr and tip chord ct: the average chord times the span. Use the projected planform and a consistent centreline root chord.
Calculator
- Mean geometric chord, (cr + ct)/21.5 m
- Taper ratio, ct/cr0.5
- Aspect ratio, b²/S8
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The wing has a planform area of 18 m², a mean chord of 1.5 m and an aspect ratio of 8.
- A rectangular wing with the root chord and the same span would have 24 m², 75 % of that, which shows the effect of taper.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Full span | 2 to 80 m | 3 | 120 | 117 | +10% | +10 % |
| ct Tip chord | 0.1 to 10 m | 12.6 | 72 | 59.4 | +10% | +3.33 % |
| cr Root chord | 0.5 to 10 m | 9 | 66 | 57 | +10% | +6.67 % |
Sweep of Tip chord
| Tip chord (m) | Planform area (m²) |
|---|---|
| 0.1 | 12.6 |
| 1.338 | 20.03 |
| 2.575 | 27.45 |
| 3.813 | 34.88 |
| 5.05 | 42.3 |
| 6.287 | 49.72 |
| 7.525 | 57.15 |
| 8.762 | 64.57 |
| 10 | 72 |
Sweep one variable
Variables and how each one contributes
Planform area
What the equation gives you. Everything else feeds it.
Full span
The full span, tip to tip, in m. Raises the area in direct proportion. Raising it by 10 % (to 13.2) moves the result by +10 %, from 18 to 19.8 m².
Root chord
The root chord at the centreline, in m. Raises the area in proportion to its share of the average chord. Raising it by 10 % (to 2.2) moves the result by +6.67 %, from 18 to 19.2 m².
Tip chord
The tip chord, in m. Raises the area in proportion to its share of the average chord. Raising it by 10 % (to 1.1) moves the result by +3.33 %, from 18 to 18.6 m².
Worked example with the values above
- Average chord: (cr + ct)/2 = (2 + 1) / 2 = 1.5 m.
- Span: b = 12 m.
- S = b × average chord = 18 m².
Assumptions and limits
- A symmetric, straight, linearly tapered wing with a planform area measured as a projected area.
- For the lift slope: attached subsonic flow, small angles and a lifting-line model.
Common mistakes
- Using the root chord instead of the mean chord to find aspect ratio.
- Mixing up semi-span and full span.
- Using the mean geometric chord to normalise pitching moment instead of the mean aerodynamic chord.
Statistics and uncertainty
Planform geometry is exact, and the aerodynamic estimate is not: lifting-line theory predicts lift-curve slope to about 5 % for unswept wings of aspect ratio above 4, and it overpredicts for low aspect ratio, swept and highly tapered wings.
Where it is used
- Setting reference geometry for coefficients, and predicting how lift grows with angle of attack.
- Locating the aerodynamic centre and the centre of gravity limits.
Maths you need
- Taper ratio: λ = tip chord divided by root chord
- Aspect ratio: AR = b²/S
- Lift-curve slope: the rate of change of lift coefficient with angle of attack