Mathematics Question and Solution
Let the side length of the regular hexagon be 1 unit.
Calculating the side length of the regular pentagon.
Let it be b.
(2/sin72) = (a/sin60)
a = 1.82118599462 units.
(1.82118599462/sin108) = (b/sin36)
b = 1.12555484451 units.
Notice; b is the side length of the regular pentagon.
d = 180-60-72
d = 48°
(2/sin72) = (e/sin48)
e = 1.56277742226 units.
f² = 1.56277742226²+1.12555484451²-2*1.56277742226*1.12555484451cos144
f = 2.56032328694 units.
(2.56032328694/sin144) = (1.56277742226/sing)
g = 21.02492451032°
h = 180-144-21.02492451032
h = 14.97507548968°
j = 108-21.02492451032
j = 86.97507548968°
k = 90-86.97507548968
k = 3.02492451032°
l = 3.02492451032+14.97507548968
l = 18°
m = 180-18-36
m = 126°
(1.56277742226/sin126) = (n/sin36)
n = 1.13542425908 units.
It implies;
Area Orange is;
0.5*1.13542425908*1.56277742226sin18
= 0.27416225634 square units.
Calculating Area Green.
It is;
0.5*2*1.56277742226sin60-0.5*1.12555484451*1.12555484451sin108-Area Orange
= 1.35340494814-0.60243424766-0.27416225634
= 0.47680844414 square units.
It implies;
Area Orange ÷ Area Green to 2 decimal places is;
0.27416225634÷0.47680844414
= 0.57499454909
≈ 0.57
