The regular hexagon is the regular polygon with six sides, as illustrated above.
The inradius 
, circumradius 
, sagitta 
, and area 
of a regular hexagon can be computed directly from the formulas
for a general regular polygon with side length
and 
sides,
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(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
Therefore, for a regular hexagon,
 |
(9)
|
so
 |
(10)
|
In proposition IV.15, Euclid showed how to inscribe a regular hexagon in a circle. To construct a regular hexagon with a compass and straightedge, draw an initial circle 
. Picking any point on the circle as the center, draw another
circle 
of the same radius. From the two points of intersection, draw circles 
and 
. Finally, draw 
centered on the intersection of circles 
and 
. The six circle-circle intersections then determine the vertices
of a regular hexagon.
A plane perpendicular to a 
axis of a cube (Gardner 1960; Holden 1991, p. 23),
octahedron (Holden 1991, pp. 22-23), and dodecahedron
(Holden 1991, pp. 26-27) cut these solids in a regular hexagonal cross
section. For the cube, the plane
passes through the midpoints of opposite sides (Steinhaus
1999, p. 170; Cundy and Rollett 1989, p. 157; Holden 1991, pp. 22-23).
Since there are four such axes for the cube and octahedron,
there are four possible hexagonal cross sections.
A hexagon is also obtained when the cube is viewed from above a corner along the
extension of a space diagonal (Steinhaus 1999, p. 170).
Take seven circles and close-pack them together in a hexagonal arrangement. The perimeter obtained by wrapping a band
around the circle then consists of six straight segments
of length 
(where 
is the diameter) and 6 arcs, each with length 
of a circle. The perimeter
is therefore
 |
(11)
|
Two intersecting congruent polygon diagonals that are not polygon diameters can divide a regular
hexagon into four regions with areas equal to 
, 
, 
, and 
of the total area. Inoue and Watanabe
(2026) proved that this is the only configuration of a regular
polygon cut by two intersecting congruent non-diameter polygon
diagonals for which all four area fractions are rational.
The intersections are in the interiors of the polygon
diagonals, with four distinct endpoints. Inoue and Watanabe (2026) acknowledge
ChatGPT, Claude, and Gemini assistance with exploring configurations, proof strategies,
and checks, and state that they independently verified the arguments.
