Hexagon
A six-sided Euclid 
showed how to inscribe a regular hexagon in aCircle. The Inradius 
, Circumradius 
, and Area 
can be computed directly from the formulasfor 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 thesolid 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 andRollett 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 obtainedby wrapping a band around the Circle then consists of six straight segments of length 
(where isthe 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.
- Fiber
- Fiber Bundle
- Fiber Space
- Fibonacci Dual Theorem
- Fibonacci Hyperbolic Cosine
- Fibonacci Hyperbolic Cotangent
- Fibonacci Hyperbolic Sine
- Fibonacci Hyperbolic Tangent
- Fibonacci Identity
- Fibonacci Matrix
- Fibonacci n-Step Number
- Fibonacci Number
- Fibonacci Polynomial
- Fibonacci Pseudoprime
- Fibonacci Q-Matrix
- Fibonacci Sequence
- Fibration
- Field
- Field Axioms
- Field Extension
- Fields Medal
- Fifteen Theorem
- Figurate Number
- Figurate Number Triangle
- Figure Eight Knot