← Louis Gagnon : Blog - airfoil center

Finding the area and center of mass of NACA 4 number symmetric airfoil
The equation for NACA 4 digit symmetric airfoils isref
$\frac{1}{20}\phantom{\rule{thinmathspace}{0ex}}\left(-0.103600000000000\phantom{\rule{thinmathspace}{0ex}}{r}_{\mathit{x}\mathit{c}}^{4}+0.284300000000000\phantom{\rule{thinmathspace}{0ex}}{r}_{\mathit{x}\mathit{c}}^{3}-0.351600000000000\phantom{\rule{thinmathspace}{0ex}}{r}_{\mathit{x}\mathit{c}}^{2}-0.126000000000000\phantom{\rule{thinmathspace}{0ex}}{r}_{\mathit{x}\mathit{c}}+0.296900000000000\phantom{\rule{thinmathspace}{0ex}}\sqrt{{r}_{\mathit{x}\mathit{c}}}\right){N}_{n}c$
where rxc is x/c or the ratio of position x to chord c and Nn is the NACA number or thickness ratio; by multiplying by two and integrating with respect to rxc going from 0 to 1 we obtain its area
$0.00680883333333\phantom{\rule{thinmathspace}{0ex}}{N}_{n}c$
and integrating the same equation from 0 to xcg and making it equal to the area divided by two we obtain the equation for the center of mass along the chord
$\frac{1}{10}\phantom{\rule{thinmathspace}{0ex}}\left(-0.02072\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{5}+0.071075\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{4}-0.1172\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{3}-0.063\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{2}+0.197933333333\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{1.5}\right){N}_{n}c-0.00340441666667\phantom{\rule{thinmathspace}{0ex}}{N}_{n}c=0$
from which Nn and c can be factored out
$-0.002072\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{5}+0.0071075\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{4}-0.01172\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{3}-0.0063\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{2}+0.0197933333333\phantom{\rule{thinmathspace}{0ex}}{x}_{\mathit{c}\mathit{g}}^{1.5}-0.00340441666667=0$
and xcg can be found numerically to give 0.3994c for a closed trailing edge, as on the equation shown, and 0.4015c for an open trailing edge