Mathematics Question and Solution
Let the side length of the regular decagon be 1 unit.
a = (180*8)/10
a = 144°
a is a interior angle of the regular decagon.
b = 144-120
b = 24°
c² = 2-2cos24
c = 0.41582338164 units.
d = ½(180-24)
d = 78°
e = 360-120-90-78
e = 72°
f = 144-78
f = 66°
g = 180-66-72
g = 42°
(0.41582338164/sin42) = (h/sin72)
h = 0.59102293779 units.
Area Orange is;
0.5*1*1sin24+0.5*0.59102293779*0.41582338164sin66
= 0.20336832154+0.11225699415
= 0.31562531569 square units.
Calculating Purple Area.
j = 180-108-54
j = 18°
(k/sin108) = (1/sin18)
k = 3.07768353718 units.
l² = 2-2cos120
l = √(3) units.
l = 1.73205080757 units.
m = 3.07768353718-1.73205080757
m = 1.34563272961 units.
n = 180-108-18
= 54°
(o/sin108) = (1/sin54)
o = 1.17557050458 units.
p = 1.34563272961-1.17557050458
p = 0.17006222503
tan54 = q/0.17006222503
q = 0.23407057189 units.
r = 1-2*0.23407057189
r = 0.53185885622 units.
(0.53185885622/sin108) = (s/sin36)
s = 0.32870685036 units.
It implies;
Area Purple is;
0.5*0.32870685036*0.53185885622sin36
= 0.05137996924 square units.
Therefore;
Area Orange ÷ Area Purple to 3 decimal places is;
0.31562531569÷0.05137996924
= 6.14296427886
≈ 6.14
