= 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> gives
$$
S=\frac{bc_r(1+\lambda)}2,\qquad\overline c=\frac{2c_r}{3}\frac{1+\lambda+\lambda^2}{1+\lambda}.
$$
A 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 station
$$
y_{\rm MAC}=\frac{2}{S}\int_0^{b/2}yc(y)\,dy=\frac{b}{6}\frac{1+2\lambda}{1+\lambda}.
$$
For a straight <leading edge> with $x_{\rm LE}(y)=x_0+y\tan\Lambda_{\rm LE}$, its corresponding chord reference starts at $x_{\rm LE,MAC}=x_0+y_{\rm MAC}\tan\Lambda_{\rm LE}$. A geometric quarter-chord reference is then $x_{1/4}=x_{\rm LE,MAC}+\overline c/4$. These are the standard planform integrals, not a guarantee about the <aerodynamic center> of every finite wing.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, equations (A-89), (A-91)–(A-92), and trapezoidal-planform relations]{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 $y_{\rm MAC}=20/9\,\mathrm m$. With $x_0=0.10\,\mathrm m$ and $\Lambda_{\rm LE}=20\,{}^\circ$, the leading-edge reference is $0.908823\,\mathrm m$ aft of the chosen origin, and \b[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.
\Image[/trapezoidal-wing-reference-chord.png]
{title=Identical trapezoidal chord distributions with different leading-edge sweep: the mean chord length stays fixed while its geometric reference position moves. Original calculation.}
The displayed $y_{\rm MAC}$ 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.
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