Editing Hexagon (section)

=== Measurement ===
The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a [[triangle]] with a vertex at the center of the regular hexagon and sharing one side with the hexagon is [[equilateral triangle|equilateral]], and that the regular hexagon can be partitioned into six equilateral triangles.

[[Image:Regular hexagon 1.svg|thumb|''R'' = [[Circumradius]]; ''r'' = [[Inradius]]; ''t'' = side length]]

The maximal [[diameter#Polygons|diameter]] (which corresponds to the long [[diagonal]] of the hexagon), ''D'', is twice the maximal radius or [[circumradius]], ''R'', which equals the side length, ''t''. The minimal diameter or the diameter of the [[inscribed]] circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), ''d'', is twice the minimal radius or [[inradius]], ''r''. The maxima and minima are related by the same factor:
:<math>\frac{1}{2}d = r = \cos(30^\circ) R = \frac{\sqrt{3}}{2} R = \frac{\sqrt{3}}{2} t</math> &nbsp; and, similarly, <math>d = \frac{\sqrt{3}}{2} D.</math>

The area of a regular hexagon
:<math>\begin{align}
  A &= \frac{3\sqrt{3}}{2}R^2 = 3Rr           = 2\sqrt{3} r^2 \\[3pt]
    &= \frac{3\sqrt{3}}{8}D^2 = \frac{3}{4}Dd = \frac{\sqrt{3}}{2} d^2 \\[3pt]
    &\approx 2.598 R^2 \approx 3.464 r^2\\
    &\approx 0.6495 D^2 \approx 0.866 d^2.
\end{align}</math>

For any regular [[polygon]], the area can also be expressed in terms of the [[apothem]] ''a'' and the perimeter ''p''. For the regular hexagon these are given by ''a'' = ''r'', and ''p''<math>{} = 6R = 4r\sqrt{3}</math>, so
:<math>\begin{align}
  A &= \frac{ap}{2} \\
    &= \frac{r \cdot 4r\sqrt{3}}{2} = 2r^2\sqrt{3} \\
    &\approx 3.464 r^2.
\end{align}</math>

The regular hexagon fills the fraction <math>\tfrac{3\sqrt{3}}{2\pi} \approx 0.8270</math> of its [[circumscribed circle]].

If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then {{nowrap|PE + PF {{=}} PA + PB + PC + PD}}.

It follows from the ratio of [[circumradius]] to [[inradius]] that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long [[diagonal]] of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides.