= Mean geometric chord
{title2=$S/b$}
= Standard mean chord
{synonym}
The <mean geometric chord>, also called the <standard mean chord>, is the span-average <airfoil chord> of an <aircraft wing>:
$$
\overline c_g=\frac{1}{b}\int_{-b/2}^{b/2}c(y)\,dy=\frac{S}{b},
$$
where $S$ is the <wing area> and $b$ the <wingspan>. The area is the projected <wing planform>, not the sum of the upper and lower skin areas.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, equations (A-88) and (A-94)]{ref}
A useful exact identity explains its difference from the <mean aerodynamic chord>. Represent uniform spanwise averaging by a coordinate $Y$ with a <continuous uniform distribution> on $[-b/2,b/2]$, and put $C=c(Y)$. This is a mathematical averaging device, not a claim that the wing geometry is random. With $\mu=\mathbb E[C]=S/b>0$,
$$
\overline c_a=\frac{\mathbb E[C^2]}{\mathbb E[C]}=\mu+\frac{\operatorname{Var}(C)}{\mu},\qquad\boxed{\frac{\overline c_a-\overline c_g}{\overline c_g}=\operatorname{CV}(C)^2.}
$$
Thus the relative excess is the squared <coefficient of variation> of the local chords. The <mean aerodynamic chord> can also be interpreted as the <expected value> under the corresponding <size-biased distribution>, since larger chords receive proportionally larger weights.
For the original linear taper from $2$ to $1\,\mathrm m$, uniform spanwise sampling makes $C$ uniform on $[1,2]\,\mathrm m$. Its <expected value> is $1.5\,\mathrm m$ and <variance> is $1/12\,\mathrm{m^2}$. Hence $\overline c_a=1.5+(1/12)/1.5=1.5556\,\mathrm m$, a relative excess of $1/27\simeq3.70\%$. A constant-chord wing has zero <variance>, so its two averages coincide.
Equal $S/b$ does not determine how local chord varies or where the wing lies relative to a reference axis. Those additional data enter <wing sweep> and the <mean aerodynamic chord reference position>.
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