= Solution
With the <graphene> lattice-vector length $a=0.246\ \mathrm{nm}$, the <carbon nanotube chiral vector> is $\mathbf C_h=n\mathbf a_1+m\mathbf a_2$. The vectors meet at $60^\circ$, giving
$$
C=a\sqrt{n^2+nm+m^2},\qquad d=\frac C\pi,\qquad
\theta=\tan^{-1}\!\left(\frac{\sqrt3\,m}{2n+m}\right).
$$
Here $C$ is circumference, $d$ diameter, and the <carbon nanotube chiral angle> $\theta$ is measured from the zigzag direction. <Nanotube zone folding> gives a nominal metallic tube when $n-m\equiv0\pmod3$ and a semiconducting tube otherwise. Evaluating all the supplied indices gives the following derived results:
$$
\begin{array}{c|rrr|c}
(n,m)&d\ (\mathrm{nm})&C\ (\mathrm{nm})&\theta\ ({}^\circ)&\text{type}\\\hline
(5,4)&0.6116&1.9213&26.33&\mathrm S\\
(6,4)&0.6826&2.1446&23.41&\mathrm S\\
(7,3)&0.6960&2.1865&17.00&\mathrm S\\
(6,5)&0.7470&2.3467&27.00&\mathrm S\\
(8,3)&0.7712&2.4228&15.30&\mathrm S\\
(10,0)&0.7830&2.4600&0.00&\mathrm S\\
(9,2)&0.7947&2.4966&9.83&\mathrm S\\
(7,5)&0.8175&2.5683&24.50&\mathrm S\\
(8,4)&0.8287&2.6034&19.11&\mathrm S\\
(7,6)&0.8824&2.7723&27.46&\mathrm S\\
(11,1)&0.9030&2.8370&4.31&\mathrm S
\end{array}
$$
\b[All eleven listed tubes are semiconducting in the zone-folding classification.] Curvature can open small gaps in some nominally metallic non-armchair tubes, but none of these indices falls in that nominal metallic class.
Diameter gives a useful inverse-size trend in transition energies, and the full chiral indices constrain the <electronic band structure> much more strongly. They do not by themselves give the perceived solution colour through a simple geometrical rule: one needs the visible <absorption spectrum>, transition strengths and widths, <exciton>/environment shifts, <concentration>, optical path and illumination. The observed transmitted colour is complementary to the absorbed spectrum, not a direct label for the <band gap>. Structure-assigned excitation–emission measurements provide an experimental connection between chiral structure and optical transitions.<Structure-assigned optical spectra of single-walled carbon nanotubes by Sergei M. Bachilo and others>[Science 298 (2002), pp. 2361–2366, structure assignment of semiconducting-tube optical transitions]{ref}
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