Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the small regular hexagon be 2 units.
a² = 2²+1²-2*2*1cos120
a = 2.64575131106 units.
Where a is the side length of the big regular hexagon.
(2.64575131106/sin120) = (1/sinb)
b = 19.1066053509°
c = 180-120-19.1066053509
c = 40.8933946491°
(2.64575131106/sin40.8933946491) = (d/sin120)
d = 3.5 units.
Area Green is;
Area triangle with height 2.64575131106 units and base 3.5sin19.1066053509 units.
= 0.5*2.64575131106*3.5sin19.1066053509
= 1.51554445662 square units.
e² = 2²+2.64575131106²-4*2.64575131106cos19.1066053509
e = 1 unit.
(1/sin19.1066053509) = (2/sinf)
f = 40.89339464921°
g = 120-f
g = 79.10660535079°
h² = 1²+2.64575131106²-2*2.64575131106cos79.10660535079
h = 2.64575131106 units.
i = 120-79.10660535079
i = 40.89339464921°
j = 79.10660535079°
(2/sin79.10660535079) = (k/sin40.89339464921)
k = 1.33333333334 units.
Area Yellow is;
Area triangle with height 1.33333333334 units and base 2sin60 units.
= 0.5*1.33333333334*2sin60
= 1.15470053839 square units.
Therefore:
Area Yellow ÷ Area Green to 2 decimal places is;
1.15470053839 ÷ 1.51554445662
= 0.76190476191
≈ 0.76
