Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the bigger regular pentagon side length be 1 unit.
a = √(2-2cos180)
a = 1.61803398875 units.
b = sin36/sin108
b = 0.61803398875 units.
c = √(2*0.61803398875²-2*0.61803398875²cos108)
c = 1 unit.
It implies;
Area Blue is;
Area triangle with height 1.61803398875 units and base sin72 units - Area triangle with height 1.61803398875 units and base 0.61803398875sin36.
= ½*1.61803398875*sin72 - ½*1.61803398875*0.61803398875sin36
= 0.76942088429 - 0.29389262615
= 0.47552825815 square units.
d=
√(1²+0.61803398875²-2*0.61803398875cos144)
d = 1.54336191843 units.
e = asin0.23537658903
e = 13.61382244063°
f = 72-13.61382244063-36
f = 22.38617755937°
g = 180-22.38617755937-108
g = 49.61382244063°
h = (0.61803398875sin22.38617755937/sin49.61382244063)
h = 0.30901699438 units.
i = 0.61803398875-0.30901699438
i = 0.30901699437 unit.
j = 108-49.61382244063
j = 58.38617755937°
k = (0.30901699437sin49.61382244063/sin58.38617755937)
k = 0.27639320224 unit.
It implies;
Area Green is;
Area triangle with height 0.61803398875 unit and base sin72 units - Area triangle with height 0.30901699437 unit and base 0.27639320224sin72 units.
= ½*0.61803398875sin72 - ½*0.30901699437*0.27639320224sin72
= 0.29389262615 - 0.04061496203
= 0.25327766412 square units.
Therefore;
Area Green ÷ Area Blue to 2 decimal places is;
0.25327766412 ÷ 0.47552825815
= 0.53262379213
≈ 0.53
