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For a symmetric trapezoidal wing planform, let root chord be $c_r$, tip chord $c_t=\lambda c_r$, and wingspan $b$. The wing taper ratio $\lambda$ describes the linear local chord $c(y)=c_r[1-2(1-\lambda )y/b]$ on one half-wing. Direct integration givesA rectangular planform has $\lambda =1$ and $\overline{c}=c_r$; the triangular limit $\lambda =0$ gives $\overline{c}=2c_r/3$. These are lengths, not aerodynamic-center locations.

The mean aerodynamic chord reference position requires a placement convention. On the positive half-span, define the area-weighted stationFor a straight leading edge with $x_{\mathrm{LE}}(y)=x_0+y\tan {\Lambda }_{\mathrm{LE}}$, its corresponding chord reference starts at $x_{\mathrm{LE},\mathrm{MAC}}=x_0+y_{\mathrm{MAC}}\tan {\Lambda }_{\mathrm{LE}}$. A geometric quarter-chord reference is then $x_{1/4}=x_{\mathrm{LE},\mathrm{MAC}}+\overline{c}/4$. These are the standard planform integrals, not a guarantee about the aerodynamic center of every finite wing.^[ref]

For the stipulated $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$, the values are $\overline{c}=14/9\mathrm{m}$ and $y_{\mathrm{MAC}}=20/9\mathrm{m}$. With $x_0=0.10\mathrm{m}$ and ${\Lambda }_{\mathrm{LE}}=20{}^{\circ }$, the leading-edge reference is $0.908823\mathrm{m}$ aft of the chosen origin, and the quarter-chord reference is $1.297712\mathrm{m}$. Changing the wing sweep moves these reference positions without changing $\overline{c}$ when the same spanwise chords are retained.

The displayed $y_{\mathrm{MAC}}$ belongs to one half-wing; the spanwise centroid of the whole symmetric planform is instead zero. A half-wing placement convention and a whole-wing centroid are different quantities.

A mean aerodynamic chord reference position locates the reference segment whose length is the mean aerodynamic chord. Its chordwise origin and spanwise placement are additional geometric data. The length alone does not specify a point on a wing or determine its aerodynamic center.

For a symmetric wing planform, let $c(y)$ be the local airfoil chord, $x_{\mathrm{LE}}(y)$ its leading-edge coordinate and $S=2{\int }_0^{b/2}c(y)dy$ the wing area. A standard area-weighted half-wing placement isThe definition averages leading-edge positions themselves; it does not generally substitute the average station into a nonlinear leading-edge curve.^[ref]

The quarter-chord reference of a mean aerodynamic chord then hasThe last identity follows directly from the definition of $\bar{c}$. It is a geometric area-weighted average of the local quarter-chord positions, without assuming a uniform aerodynamic loading.

For an original curved-leading-edge example, take constant chord $c=0.4\mathrm{m}$, span $b=4\mathrm{m}$ and $x_{\mathrm{LE}}(y)=k{y}^2$ on the positive half-wing, with $k=0.05{\mathrm{m}}^{-1}$. Then $S=1.6{\mathrm{m}}^2$, $\bar{c}=0.4\mathrm{m}$, $y_{\mathrm{MAC}}=1\mathrm{m}$ andEvaluating the leading edge at the mean station would instead give $x_{\mathrm{LE}}(1\mathrm{m})=0.05\mathrm{m}$, moving the quarter reference forward by $16.667\mathrm{mm}$. For a straight leading edge the two operations agree, as in mean aerodynamic chord of a trapezoidal wing.

The positive half-wing station is a placement convention; the whole symmetric planform's spanwise centroid remains zero. Reporting a chord-reference position should identify its coordinate origin, planform-area convention and the chosen fraction of the reference chord.

The mean geometric chord, also called the standard mean chord, is the span-average airfoil chord of an aircraft wing: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.^[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 $\mu =\mathbb{E}[C]=S/b>0$,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 $1/27\simeq 3.70\%$. 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.

The quarter-chord reference of a mean aerodynamic chord is the geometric point one quarter of the reference-chord length aft of its specified leading-edge position:It supplies a reproducible location for reporting positions and pitching moments. The length of the mean aerodynamic chord alone does not locate this point; its reference origin is also required.

For a symmetric wing planform using the area-weighted placement in mean aerodynamic chord reference position,Adding one quarter of the first integral to the second givesIt is consequently an area-weighted average of local quarter-chord coordinates under this convention. No aerodynamic-load distribution was needed for that identity.^[ref]

For an original trapezoidal example, let span $b=8$ m, root chord 3 m and tip chord 1 m. Then $S=16{\mathrm{m}}^2$, $\bar{c}=13/6$ m and the positive half-wing reference station is $y_{\mathrm{MAC}}=5/3$ m. With straight leading edge $x_{\mathrm{LE}}(y)=0.2\mathrm{m}+0.3y$, its reference leading edge is at 0.7 m. The quarter-chord point is therefore $x_{1/4}=1.241667$ m aft of the stated origin. Changing sweep changes this coordinate even if all spanwise chord lengths remain fixed.

The aerodynamic center is defined through the response of pitching moment to incidence, rather than through a geometric averaging rule. In ideal two-dimensional thin airfoil theory, the quarter-chord moment is independent of angle of attack, which motivates that familiar reference. It does not prove that every finite wing's aerodynamic center lies at this geometric point.^[ref]

Moment comparisons must also use the same reference. Moving the reference by a vector $\Delta \mathbf{r}$ changes the moment by $-\Delta \mathbf{r}\times \mathbf{F}$ for the same resultant force. A location convention thus has a measurable mechanical consequence even though it is chosen geometrically.