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Regular Hexagon


RegularHexagon

The regular hexagon is the regular polygon with six sides, as illustrated above.

The inradius r, circumradius R, sagitta s, and area A of a regular hexagon can be computed directly from the formulas for a general regular polygon with side length a and n=6 sides,

r=1/2acot(pi/6)
(1)
=1/2sqrt(3)a
(2)
R=1/2acsc(pi/6)
(3)
=a
(4)
s=1/2atan(pi/(12))
(5)
=1/2(2-sqrt(3))a
(6)
A=1/4na^2cot(pi/6)
(7)
=3/2sqrt(3)a^2.
(8)

Therefore, for a regular hexagon,

 R/r=sec(pi/6)=2/(sqrt(3)),
(9)

so

 (A_R)/(A_r)=(R/r)^2=4/3.
(10)
HexagonConstruction

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 A. Picking any point on the circle as the center, draw another circle B of the same radius. From the two points of intersection, draw circles C and D. Finally, draw E centered on the intersection of circles A and C. The six circle-circle intersections then determine the vertices of a regular hexagon.

CubeHexagon
DodecahedronHexagon
OctahedronHexagon

A plane perpendicular to a C_3 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).

CirclesHexagonal

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 d (where d is the diameter) and 6 arcs, each with length 1/6 of a circle. The perimeter is therefore

 p=(12+2pi)r=2(6+pi)r.
(11)

See also

Centroid Hexagon, Cosine Hexagon, Cube, Cyclic Hexagon, Dissection, Dodecahedron, Graham's Biggest Little Hexagon, Heptagon Theorem, Hexagon, Hexagon Polyiamond, Hexagram, Lemoine Hexagon, Magic Hexagon, Octahedron, Pappus's Hexagon Theorem, Pascal's Theorem, Permutohedron, Polygon, Regular Polygon, Talisman Hexagon, Tucker Hexagon

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References

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.

Cite this as:

Weisstein, Eric W. "Regular Hexagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularHexagon.html

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