= Mean aerodynamic chord
{title2=$\bar c$}
The \b[mean aerodynamic chord] is a reference <length> that weights each local <airfoil chord> by the <area> of its spanwise strip. For a symmetric <aircraft wing> with <wingspan> $b$, local chord $c(y)>0$ and <wing area> $S$, its geometric definition is
$$
S=2\int_0^{b/2}c(y)\,dy,\qquad\bar c=\frac{2}{S}\int_0^{b/2}c(y)^2\,dy.
$$
The extra factor of $c(y)$ distinguishes this from the <mean geometric chord> $S/b$.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, pp. A-27–A-28, equations (A-88), (A-89) and (A-94)]{ref}
The weighting has a simple interpretation: a strip has <area> $dS=c(y)\,dy$, so $\bar c=(1/S)\int c\,dS$ is an <area>-weighted <mean>. If a simplified section model has the same local <pitching moment coefficient> $c_m$ and <dynamic pressure> $q$ everywhere, its strip moment is $dM=qc_m c(y)^2\,dy$. Integration then gives $M=qc_mS\bar c$. This explains its use as a moment reference; an actual three-dimensional flow need not have a uniform section coefficient.
For the <mean aerodynamic chord of a trapezoidal wing>, take root chord $c_r$, tip chord $c_t=\lambda c_r$ and a linear distribution $c(y)=c_r[1-2(1-\lambda)y/b]$. Integrating gives
$$
S=\frac{bc_r(1+\lambda)}2,\qquad\boxed{\bar c=\frac{2c_r}{3}\frac{1+\lambda+\lambda^2}{1+\lambda}.}
$$
The stipulated <wing planform> $b=10\,\mathrm m$, $c_r=2\,\mathrm m$, $c_t=1\,\mathrm m$ has $S=15\,\mathrm{m^2}$, $S/b=1.5\,\mathrm m$ and \b[$\bar c=14/9\,\mathrm m\simeq1.5556\,\mathrm m$]. The wider strips receive more weight.
\Image[/mean-aerodynamic-chord.png]
{title=Trapezoidal wing planform with a mean aerodynamic chord of 1.5556 m, compared with the span-average chord of 1.5 m. Original geometry calculation.}
The <Cauchy-Schwarz inequality> gives a useful general check:
$$
\left(\int_{-b/2}^{b/2}c\,dy\right)^2\le b\int_{-b/2}^{b/2}c^2\,dy,\qquad\bar c\ge S/b.
$$
Equality holds for a constant chord almost everywhere. A reported <mean aerodynamic chord> shorter than $S/b$ therefore contradicts this definition for the same span and area convention. Its <length> alone does not specify a location: the reference position of that chord and the <aerodynamic center> are additional geometric or aerodynamic data.
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