= Wing aspect ratio
{title2=$AR=b^2/S$}
The \b[wing aspect ratio] is the dimensionless measure
$$
AR=\frac{b^2}{S},
$$
where $b$ is <wingspan> and $S$ is projected <wing area>. For a rectangular <wing planform>, $S=bc$ and therefore $AR=b/c$, where $c$ is <airfoil chord>. For other <wing planforms>, dividing the span by a chosen root chord need not give the <wing aspect ratio>.<Wing Geometry by NASA Glenn Research Center>[Top View, aspect ratio and reference area]{ref}
Its aerodynamic importance comes from <lift-induced drag>. The <wingtip vortices> of a finite <aircraft wing> change the local flow direction; maintaining <lift> then entails an additional rearward force. A standard model writes
$$
C_{D_i}=\frac{C_L^2}{\pi e\,AR},\qquad D_i=qS C_{D_i}=\boxed{\frac{L^2}{\pi e q b^2}},\qquad q=\frac12\rho V^2.
$$
Here $e$ is a <span efficiency factor>, $q$ is <dynamic pressure> and $C_L=L/(qS)$ is <lift coefficient>. The model compares a specified spanwise loading in a regime where this finite-wing approximation is appropriate; $e$ need not stay unchanged when the design changes.<Induced Drag Coefficient by NASA Glenn Research Center>[Induced Drag Coefficient, aspect ratio and efficiency]{ref}
For an original example, take $S=16\,\mathrm{m^2}$, $q=500\,\mathrm{Pa}$, $C_L=0.6$ and $e=0.8$. Both wings then supply $L=4800\,\mathrm N$. Spans of 8 and 12 metres give $AR=4$ and $9$, respectively, and <lift-induced drag> approximately $286.48$ and $127.32\,\mathrm N$. Under those stipulated conditions, increasing span by 50% reduces that drag component by $1-(8/12)^2=5/9$, about 55.6%.
\Image[/wing-aspect-ratio-induced-drag.png]
{title=Equal-area wings with different span, and the inverse-square induced-drag comparison at fixed lift, dynamic pressure and span efficiency. Original model calculation.}
The boxed expression also exposes a common comparison error. At fixed $L$, $q$, $e$ and $b$, changing only the reference <wing area> changes both <wing aspect ratio> and <lift coefficient>; their effects cancel in this model's dimensional <lift-induced drag>. A larger numerical $AR$ is therefore not, by itself, a promise of smaller total <drag>. Structural loads, <wing loading>, surface losses and <aerodynamic stall> impose further design constraints.
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