79. To cut off a regular hexagon from a given square.

Fig. 29.

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.

Fig. 30.

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.

Fig. 31.

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.