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Does Wing Area vary with angle ot attack?
Edit: why we don't recognize that the projected area of the wing as variable (as projected on to a surface that is perpendicular to the relative wind)
Per the above link, for lift and drag equations the area is defined to the reference area by convention - it does not change. Why not have a variable projected area instead? Wouldn’t using projected area tell you the true values for lift and drag?
The convention of keeping area constant emerged in an age before computers. For efficiency, it was agreed to keep area constant and put all AoA-related changes into the coefficients. Now you need to know that the lift and drag equations are only approximations, valid only for small angles. Actually, the lift equation in full would be a Taylor series, the first element of which is $$L = \frac{\rho}{2}\cdot v^2 \cdot A \cdot c_{L\alpha} \cdot sin(\alpha) \cdot cos(\alpha)$$ To simplify things, it became customary to use $\alpha$ in radians instead of its sine and to use 1 for the cosine. Keeping the cosine would in effect give you the projected chord already but increases the computational effort. This simplification is fine for the usual range of angles of attack and the error incurred by this is negligible. However, at 90° angle of attack you will clearly see a difference and only the trigonometric functions will make the result of the equation look at least somewhat close to reality. In addition, you now need two different $c_{L\alpha}$s to get realistic results, one for the AoA regime of attached flow and one for separated flow.
Lift coefficient over the first 180°, taken from Hoerner's Fluid Dynamic Lift.
Why not have a variable projected area instead?
For simplicity in an age of slide rules. With computers, that simplification is no longer needed.
Wouldn’t using projected area tell you the true values for lift and drag?
Yes, but the difference only becomes significant when separation makes the linear equations invalid anyway.
Wouldn’t using projected area tell you the true values for lift and drag?
No. For a given airspeed and angle-of-attack, a wing makes the same amount of lift, and drag, regardless of the bank angle. The "projected area" is irrelevant.
Otherwise, all you'd have to do is give a wing lots of dihedral, and it would automatically tend to roll toward wings-level whenever it was banked, regardless of whether there was any sideslip or not. Because the lower wing would have more "projected area" than the higher wing.
That's not how it works. The stabilizing effect of dihedral is intimately connected with sideslip, which increases the angle-of-attack of the low ("upwind") wing and decreases the angle-of-attack of the high ("downwind") wing. Take away the sideslip, and the stabilizing effect of dihedral vanishes.
That's why increasing the vertical fin area can promote spiral instability-- because it reduces the tendency to sideslip while turning.
The tendency to sideslip while turning may be so slight as to be difficult to perceive, but it always there.
If you don't believe that the stabilizing effect of dihedral is dependent on sideslip-- if you are instead a believer in the (faulty) "projected area" theory-- then try using the rudder to command a slight amount of skid (rudder toward low wingtip, ball toward high wingtip), rather than slip, while banked and turning, while keeping the ailerons centered. No matter how much dihedral your plane may have, it won't roll the plane back to wings-level if you are not allowing the plane to sideslip at least a little bit as it turns.
I know this is only current because of a bot, but why not, it is here anyway.
In principle, you could make up a new "alien" aerodynamics, where you do use projected area. The reason you can do this, is because the coefficient of lift and drag are non-dimensionalised parameters, like Mach number or Reynolds number. If we consider just 2D flow, so an aerofoil, then $$c_l = \frac{L’}{0.5 \rho v^2 c}$$ Here you could choose to make c (the chord length) the projected chord length instead. This would be a shorter length than the geometric chord length and for the equation to be consistent, you would simply need to make $c_l$ larger. The factor would be by the same $\cos{\alpha}$ amount. This would then carry over into 3D lift, given the constituent aerofoils have a greater coefficient of lift, this would be offset by the smaller projected area.
So the key feature here is that the chord is used as an indicative length measure for the non-dimensionalising. You could just have easily used the quarter chord dimension (c/4), but then all coefficients of lift would just be 4 times larger. Obviously, using the geometric c makes things easier, because the "projected effect" is hidden in the current values we used for $c_l$. That is, the system we have now makes things easier, such that we don't need to have a $c \cos{\alpha}$ or an $S \cos{\alpha}$ (for surface area), whenever we use them.
The projected wing area relative to the airflow does indeed vary with AoA, but we don't use projected area relative to airflow around a wing for the lift & drag equations. The reference area is chosen independent of the direction of the flow - variation with flow direction is accounted for in the definition of lift coefficient C$_L$ and drag coefficient C$_D$, not in the wing area.
The airflow around an aeroplane is such that the wetted wing area would be most relevant, as stated in this answer.. But this area changes per wing profile shape - in order to obtain an independent variable, the wing area is chosen as a flat area in its widest local chord. There are standard methods defined, like stated in this answer
Area is held constant in the Lift Equation unless you change wing configuration in flight.
Lift = Air density × Wing Area × Lift Coefficient × Airspeed$^2$
Lift Coefficient is determined by airfoil type and Angle of Attack.
As we can see, only AoA and Airspeed are variable when determining Lift and Drag in a given working fluid (air, water).
Regarding the effect of area on the calculated result, we are saying (approximately, discounting wing root and tip effects) that Lift is directly proportional to an increase in span. Changing chord would change the airfoil.
In real life, lift and drag are increased by lowering flaps, which increases Coefficient of Lift (therefor Lift). This allows the plane to fly slower at the same AoA.
