Mathematics Question and Solution
Let the side length of the inscribed yellow square be 1 unit.
Area yellow square is;
1²
= 1 square unit.
Side length of the small regular pentagon is 1 unit.
a² = 2-2cos108
a = 1.61803398875 units.
a is the side length of the big regular pentagon.
b² = 1.61803398875²+1.61803398875²-2*1.61803398875*1.61803398875cos108
b = 2.61803398875 units.
c = ½(2.61803398875-1)
c = 0.80901699438 units.
cos36 = 0.80901699438/e
e = 1 unit
sin36 = f/1
f = 0.58778525229 units.
g = 1.61803398875-1
g = 0.61803398875 units.
sin72 = h/1
h = 0.9510565163 units.
j² = 2.61803398875²-0.80901699438²
j = 2.48989828488 units.
sin36 = k/0.61803398875
k = 0.363271264 units.
l = 2.48989828488-0.9510565163-1-0.363271264
l = 0.17557050458 units.
m = 180-2*54
m = 72°
(0.17557050458/sin72) = (n/sin54)
n = 0.14934919164 units.
Area Purple is;
(0.5*0.14934919164*0.14934919164sin72)+(1*0.17557050458)+(0.5*0.61803398875*0.61803398875cos108)
= 0.01060674389+0.17557050458+0.181635632
= 0.36781288047 square units.
Therefore;
Area Purple ÷ Area Yellow to 2 decimal places is;
0.36781288047÷1
= 0.36781288047
≈ 0.37
