Mathematics Question and Solution
Let the side of the regular octagon be 1 unit.
Calculating Area Yellow.
a = ⅛(180*6)-90
a = 45°
b² = 2-2cos45
b = 0.7653668647 units.
c = ½(180-45)
c = 67.5°
d = 135-c
d = 135-67.5
d = 67.5°
It implies;
Area Yellow is;
½*sin45 + ½*0.7653668647sin67.5
= ¼(√(2))+0.3535533906
= 0.7071067812 square units.
= ½(√(2)) square units.
Calculating Area Blue.
e² = 0.7653668647²+1²-2*0.7653668647cos67.5
e = 1 unit.
f = 360-135-90
f = 135°
g = ½(180-135)
g = 22.5°
(1/sin22.5) = (h/sin135)
h = 1.847759065 units.
j = 90-22.5
j = 67.5°
cos67.5 = k/1.847759065
k = 0.7071067812 units
k = ½(√(2)) units.
l = 1-k
l = 1-½(√(2))
l = ½(2-√(2)) units.
l = 0.2928932188 units.
It implies;
Area Blue is;
0.5*0.2928932188sin135
= ½*½(2-√(2))*(√(2)/2)
= ⅛(2√(2)-2) square units.
Therefore;
Area Blue ÷ Area Yellow exactly is;
⅛(2√(2)-2)÷½(√(2))
= (√(2)-1)/(2√(2))
= (2-√(2))/4
= ¼(2-√(2))
