A six-sided Polygon. In proposition IV.15, Euclid showed how to inscribe a regular hexagon in a
Circle. The Inradius , Circumradius , and Area can be computed directly from the formulas
for a general regular Polygon with side length and sides,

(1) | |||

(2) | |||

(3) |

Therefore, for a regular hexagon,

(4) |

(5) |

A Plane Perpendicular to a axis of a Cube, Dodecahedron, or Icosahedron cuts the solid in a regular Hexagonal Cross-Section (Holden 1991, pp. 22-23 and 27). For the Cube, the Plane passes through the Midpoints of opposite sides (Steinhaus 1983, 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.

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 with total length of a Circle. The Perimeter is therefore

(6) |

**References**

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.

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 American ed.* New York: Oxford University Press, 1983.

© 1996-9

1999-05-25