Mathematics Question and Solution
Let the side length of the regular pentagon be 1 unit.
tan36 = a/0.5
a = 0.363271264 unit.
b² = 0.363271264²+0.5²
b = 0.61803398875 unit.
tan72 = c/0.5
c = 1.53884176859 units.
Area Green is;
0.5sin108+0.5*1.53884176859*0.5-0.5*0.5*0.363271264-0.5*0.61803398875*0.61803398875sin36
= 0.65716389015 square units.
Calculating Area Blue.
Let d be the radius of the inscribed blue circle.
Calculating d.
e = 0.5(90+36)
e =63°
(d/tan63) + (d/tan45) = a
d = 0.363271264/((1/tan63)+(1/tan45))
d = 0.24065262637 unit.
Notice, d is the radius of the inscribed blue circle.
Area Blue circle is;
π(0.24065262637)²
= 0.1819412123 square units.
Calculating Area of the blue inscribed regular triangle.
f = ((0.24065262637/tan63)+(0.24065262637/tan60))
f = 0.26155949624 unit.
Let g be the side of the inscribed blue regular triangle.
(g/sin36) = (h/sin24)
h = 0.69198170844g unit.
i² = 2-2cos108
i = 1.61803398875 units.
j = i-b
j = 1 unit.
Blue inscribed regular triangle side is;
0.26155949624+g+0.69198170844g = 1
1.69198170844g = 0.73844050376
g = 0.43643527591 unit.
Area Blue regular triangle is;
0.5*0.43643527591*0.43643527591sin(60)
= 0.08247841918 square units.
Area Blue Total is;
0.08247841918+0.1819412123
= 0.26441963148 square units.
It implies;
Area Blue ÷ Area Green to 2 decimal places is;
0.26441963148÷0.65716389015
= 0.40236482169
≈ 0.40
