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,
 |  |  |
(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)
|
