Mathematics Question and Solution
Let the side length of the regular hexagon be 1 unit.
a² = 1²+1²
a = √(2) units.
b = 180-120
b = 60°
c = 60+45
c = 105°
d² = 2+1-2√(2)cos105
d = 1.9318516526 units.
(1.9318516526/sin105) = (√(2)/sine)
e = 45°
f = 180-60-45
f = 75°
(1/sin75) = (g/sin60)
g = 0.8965754722 units.
It implies;
Area Green is;
0.5*0.8965754722sin45
= 0.3169872981 square units.
= (sin60sin45)/2sin75 square units.
h = ½(180-150)
h = 15°
j = 90-15
j = 75°
k = 180-120
k = 60°
(1/sin60) = (l/sin45)
l = 0.8164965809 units.
It implies;
Area Orange is;
0.5*0.8164965809sin75
= 0.3943375673 square units.
= (sin45sin75)/2sin60 square units.
Calculate Red Area.
tan30 = m/1
m = ⅓(√(3)) units.
Therefore;
Area Red is;
0.5*(1/√(3))
= ⅙(√(3)) square units.
= 0.2886751346 square units.
= ½(tan30) square units.
= (sin30)/2cos30 square units.
It implies;
Area Red : Area Green: Area Orange in the form 1:a:b, where a and b are exact fractions is;
(sin30)/(2cos30) : (sin60sin45)/(2sin75) : (sin45sin75)/(2sin60)
= 1 : (3√(2))/(4sin75) : (√(2)sin75)
= 1 : (3√(2)/(√(6)+√(2))) : (√(2)(√(6)+√(2))/4)
= 1 : ¼(6√(3)-6) : ¼(2√(3)+2)
= 1 : ½(3√(3)-3) : ½(√(3)+1)
