Mean aerodynamic chord of a trapezoidal wing

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.