Mathematics Question and Solution
Let the inscribed blue equilateral triangle side length be 1 unit.
Area Blue is;
0.5*1*1sin60
= ¼√(3) square units.
= 0.4330127019 square units.
0.5² = 2-2cosa
2cosa = 2-0.25
cosa = (1.75/2)
a = 28.9550243719°
b = ½(180-28.9550243719)
b = 75.5224878141°
c = 90-b
c = 14.4775121859°
sin60 = d/1
d = ½√(3) units.
d = 0.8660254038 units.
e² = 1²+0.8660254038²-2*0.8660254038cos14.4775121859
e = 0.2700907567 units.
(0.2700907567/sin14.4775121859) = (1/sinf)
f = 67.76124391°
g = 180-f
g = 112.23875609°
h = 180-g-30
h = 180-142.23875609
h = 37.76124391°
j = 180-h-120
j = 180-157.76124391
j = 22.23875609°
It implies
(1/sin22.23875609) = (k/sin37.76124391)
k = 1.6180339891 units
Area Golden is;
0.5*1.6180339891*1sin120
= 0.7006292694 square units.
Therefore;
Area Golden ÷ Area Blue is;
= 0.7006292694÷0.4330127019
= 1.618033989
≈ 1.62 to 2 decimal places.
