Suppose I want to cover a whole circle of with hexagons. See the attached photo where coloured hexagons are those required to cover a whole circle. Given the length of the side of the hexagons and the radius of the circle, what's the minimum number of hexagons required to cover such circle?
I notice that there are three ways of inscribing a circle in the hexagon tile: the center of the circle lies
- at the center of a hexagon,
- at the center of a hexagon side, and
- at the vertex between three hexagons.
This is a response to the first of the three cases noted by OP.
Circle concentric with center of hexagon
For a regular hexagon with side $s=1$, the number of hexagons needed to cover the circle increases by $6$ as radius $r$ increases just beyond points of tangency
$A$, $C$, $E$, $G$… at distances $\frac{\sqrt 3}{2}$,$\frac{3\sqrt 3}{2}$, $\frac{5\sqrt 3}{2}$, $\frac{7\sqrt 3}{2}$… from center $O$, and beyond vertices $B$, $F$, $J$… at distances $2$, $5$, $8$… from center (see figure below). As the increasing circle passes beyond remaining vertices $D$, $F’$, $H$, $H’$, $J’$, $J’’$…the number of hexagons increases by $12$. These remaining vertices become a majority with increasing $r$—since double-primed vertices appear for $J$ and $L$, triple-primed for $N$ and $Q$, etc.
All circles within these bands, (reminiscent of Saturn's rings), are covered by the same number of hexagons, and each circle shown in the figure is the greatest circle covered by that number. E.g. $OD$ is radius of the greatest circle covered by $19$ hexagons, $OE$ is radius of the greatest circle covered by $31$, and $OF$ is radius of the greatest circle covered by $37$.
This sequence of the number of hexagons needed to cover the successive bands of circles whose greatest radii are $OA$, $OB$, $OC$, $OD$, $OE$, $OF$, $OF’$, $OG$, $OH$, $OH’$, $OI$, $OJ$, $OJ’$, $OJ’’$… is $1$, $7$, $13$, $19$, $31$, $37$, $43$, $55$, $61$, $73$, $85$, $91$, $97$, $109$…, which is https://oeis.org/A038590
Any radius not in the $2$, $5$, $8$… or $\frac{\sqrt 3}{2}$,$\frac{3\sqrt 3}{2}$, $\frac{5\sqrt 3}{2}$, $\frac{7\sqrt 3}{2}$… sequences noted above is easily calculated using the Pythagorean Theorem. E.g.$$OD=\sqrt {\left(\frac{\sqrt 3}{2}\right)^2+\left(\frac{7}{2}\right)^2}=\sqrt {13}$$Given any $r$, then, and hence the band within which its circle falls, the number of hexagons required to cover that circle is that member having the same position in sequence OEIS A038590 that the band within which $r$ lies has in the sequence of bands defined successively by $OA$, $OB$, $OC$….
For example, if $r=6$, then $r$ lies between $OF’=2\sqrt 7\approx 5.29$ (Pythagoras) and $OG=\frac{7\sqrt 3}{2}\approx 6.06$. Thus $r$ lies within the eighth band, corresponding to the eighth member of OEIS A038590: $55$ hexagons (of side $s=1$) is the minimum number needed to cover a concentric circle of $r=6$, shown below just within the circle with radius $OG\approx 6.06$.