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,
Therefore, for a regular hexagon,
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(5)
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so
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(6)
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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
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(7)
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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).
Cadwell, J. H. Topics in Recreational Mathematics. Cambridge, England:
Cambridge University Press, 1966.
Coxeter, H. S. M. and Greitzer, S. L. "Hexagons." §3.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer.,
pp. 73-74, 1967.
Cundy, H. and Rollett, A. "Hexagonal Section of a Cube." §3.15.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin
Pub., p. 157, 1989.
Dixon, R. Mathographics. New York: Dover, p. 16, 1991.
Gardner, M. "Mathematical Games: More About the Shapes that Can Be Made with
Complex Dominoes." Sci. Amer. 203, 186-198, Nov. 1960.
Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.
Pappas, T. "Hexagons in Nature." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 74-75, 1989.
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, pp. 53-54, 1991.
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