This section is from the book "Geometric Exercises In Paper Folding", by Tandalam Sundara Row . Also available from Amazon: T. Sundara Row's Geometric Exercises in Paper Folding (Large Print Edition).
79. To cut off a regular hexagon from a given square.
Fold through the mid-points of the opposite sides, and obtain the lines A OB and COD.
On both sides of AO and OB describe equilateral triangles (§ 25), AOE, AHO; BFO and BOG.
Draw EF and HG.
AHGBFE is a regular hexagon.
It is unnecessary to give the proof.
The greatest breadth of the hexagon is AB.
80. The altitude of the hexagon is
√3/2. AB = 0.866.. .. X AB.
81. If R be the radius of the circumscribed circle,
R= 1/2 AB
82. If r be the radius of the inscribed circle, r = √3 / 4. AB = 0.433.. .. X AB.
83. The area of the hexagon is 6 times the area of the triangle HGO,
=6.AB/4 . √3/4 AB.
= 3√3 / 8. AB2 = 0.6495.. .. X AB2.
Also the hexagon = ¾ . AB • CD.
= 1½ times the equilateral triangle on AB.
84. Fig. 30 is an example of ornamental folding into equilateral triangles and hexagons.
85. A hexagon is formed from an equilateral triangle by folding the three corners to the center.
The side of the hexagon is 1/3 of the side of the equilateral triangle.
The area of the hexagon = 2/3 of the equilateral triangle.
86. The hexagon can be divided into equal regular hexagons and equilateral triangles as in Fig. 31 by folding through the points of trisection of the sides.