Mathematics Question and Solution
Let the regular pentagon side be 1 unit.
Let the width and length of the inscribed red rectangle be a and b respectively.
(1/sin72) = (c/sin54)
c = 0.85065080835 units.
d = 180-54-108
d = 18°
(d/sin18) = (0.85065080835/sin108)
d = 0.27639320225 units.
Calculating a.
cos18 = a/0.27639320225
a = 0.26286555606 units.
(e/sin54) = (0.85065080835/sin108)
e = 0.72360679775 units.
f = (0.72360679775-b) units.
sin72 = g/1
g = 0.9510565163 units.
h = 0.5g
h = 0.47552825815 units.
h is radius of the inscribed circle.
tan54 = j/0.5
j = 0.68819096024 units.
k = 0.68819096024-0.47552825815
k = 0.21266270209 units.
Calculating b.
0.21266270209²+(0.72360679775-b)² = 0.47552825815²
0.04522542486+0.52360679775-1.4472135955b+b² = 0.2261271243
b²-1.4472135955b+0.34270509831 = 0
Solving the quadratic equation.
b = 0.298281 units.
It implies;
Area Red Rectangle is;
ab
= 0.26286555606*0.298281
= 0.07840780093 square units.
Calculating Area Green Inscribed Circle.
l = (0.72360679775-b)
l = 0.72360679775-0.298281
l = 0.42532579775 units.
tanm = 0.42532579775/0.85065080835
m = 26.56507238454°
n = 54+26.56507238454
n = 80.56507238454°
o = 0.5*80.56507238454
o = 40.28253619227°
p = 0.5*36
p = 18°
Calculating r, radius of the inscribed green circle.
(r/tan18)+(r/tan40.28253619227) = 1
4.25757192637r = 1
r = 0.23487565619 units.
Area Green Circle is;
π(0.23487565619)²
= 0.1733109032 square units.
Therefore;
Area Red ÷ Area Green to 2 decimal places is;
0.07840780093÷0.1733109032
= 0.45241124178
≈ 0.45
