Mathematics Question and Solution
Let the side length of the regular pentagon be 1 unit.
a² = 1²+1²
a = √(2) units.
b = ½(a)
b = ½√(2) units.
c = ½(360-108-108)
c = ½(360-216)
c = ½(144)
c = 72°
d = 180-108-45
d = 72-45
d = 27°
(0.5√(2)/sin108) = (e/sin45)
e = 0.5257311121 units.
(0.5√(2)/sin108) = (f/sin27)
f = 0.3375401519 units.
sin18= g/0.3375401519
g = 0.1043056432 units.
Length AB = 2e+2g+1
AB = 2.2600735106 units.
h = 1-0.3375401519
h = 0.6624598481 units.
j = AB-e
j = 2.2600735106-2*0.5257311121
j = 1.2086112864 units.
k = 90-72
k = 18°
l = 180-108-18
l = 54°
(1.2086112864/sin54) = (m/sin18)
m = 0.4616484322 units.
n = 1-m-f
n = 1-0.4616484322-0.3375401519
n = 0.2008114159 units.
o² = 0.4616484322²+0.5257311121²-2*0.4616484322*0.5257311121cos72
o = 0.5826781348 units.
(0.5826781348/sin72) = (0.5257311121/sinp)
p = 59.1046611054°
q = 180-108-59.1046611054
q = 12.8953388946°
(0.2008114159/sin12.8953388946) = (r/sin108)
r = 0.8557705184 units.
s = r+o
s = 0.8557705184+0.5826781348
s = 1.4384486532 units.
t = 180-72-59.1046611054
t = 48.8953388946°
u = 90-48.8953388946
u = 41.1046611054°
v = 180-72-41.1046611054
v = 66.8953388946°
(0.5257311121/sin66.8953388946) = (w/sin41.1046611054)
w = 0.3757759989 units.
Area Green is;
Area triangle with height 0.5257311121sin72 units and base 0.3757759989 units.
0.5*0.3757759989*0.5257311121sin72
= 0.0939439997 units.
Area Orange is;
Area triangle with height 2.2600735106sin48.8953388946 units and base 1.4384486532 units - Area Green.
= (0.5*1.4384486532*2.2600735106sin48.8953388946)-Area Green.
= 1.2248302468-0.0939439997
= 1.1308862471 square units.
Therefore;
Area Orange ÷ Area Green to 1 decimal place is;
1.1308862471÷0.0939439997
= 12.0378762952
≈ 12
