Hexagon
A hexagon is a six-sided polygon.
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)
|
Therefore, for a regular hexagon,
 |
(5)
|
so
 |
(6)
|
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
 |
(7)
|
Given an arbitrary hexagon, take each three consecutive vertices, and mark the fourth point of the parallelogram sharing these three vertices. Taking alternate points then gives two congruent triangles, as illustrated above (Wells 1991).
Given an arbitrary hexagon, connecting the centroids of each consecutive three sides gives a hexagon with equal and parallel sides known as the centroid hexagon (Wells 1991).
Contact the MathWorld Team
hexagon
