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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(2)
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(6)
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(7)
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(8)
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Therefore, for a regular hexagon,
(9)
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so
(10)
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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
(11)
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