Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the side of the regular pentagon be 1 unit.
a = 360-2(108)-90
a = 54°
tan72 = b/0.5
b = 1.53884176859 units.
tan54 = 1.53884176859/c
c = 1.11803398875 units.
d = c-0.5
d = 0.61803398875 unit.
(e/tan36)+(e/tan27) = d
Where e is the radius of the inscribed circle.
e = 0.61803398875/((1/tan36)+(1/tan27))
e = 0.18509595408 unit.
f = 180-72-54
f = 54°
Therefore;
g = d = 0.61803398875 unit.
Shaded Area Red is;
Area triangle with height 0.61803398875 units and base 0.61803398875sin72 units - Area circle with radius 0.18509595408 units.
= 0.5*0.61803398875*0.61803398875sin72-π(0.18509595408*0.18509595408)
= 0.07400305851 square units.
Calculating Shaded Area Blue.
h = 1-d
h = 0.38196601125 unit.
Let the inscribed yellow square side be i.
Calculating i.
(i/sin54) = (j/sin54)
j = i unit.
(i/sin108) = (k/sin54)
k = 0.85065080835i
It implies;
j+k = h
i+0.85065080835i = 0.38196601125
i = 0.38196601125/(1+0.85065080835)
i = 0.20639550667 unit.
Notice;
i is the side of the inscribed square.
Shaded Area Blue is;
Area triangle with height 0.38196601125 unit and base sin108 units - Area square with side 0.20639550667 unit.
= 0.5*0.38196601125sin108-0.20639550667*0.20639550667
= 0.13903652683 square units.
Shaded Area Red ÷ Shaded Area Blue to 2 decimal places is;
0.07400305851÷0.13903652683
= 0.53225623652
≈ 0.53
