Hexagon

In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

A regular hexagon is defined as a hexagon that is both equilateral and equiangular. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of its internal angle is equal to 120°. The Schläfli symbol denotes this polygon as ${\displaystyle \{6\}}$. However, the regular hexagon can also be considered as the cutting off the vertices of an equilateral triangle, which can also be denoted as ${\displaystyle \mathrm {t} \{3\}}$.

A regular hexagon is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals ${\displaystyle {\tfrac {2}{\sqrt {3}}}}$ times the apothem (radius of the inscribed circle).

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, and that the regular hexagon can be partitioned into six equilateral triangles.

The maximal 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:

The area of a regular hexagon

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${\displaystyle {}=6R=4r{\sqrt {3}}}$, so

The regular hexagon fills the fraction ${\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}$ of its circumscribed circle.