Mathematics Question and Solution
Let the the side length of the inscribed hexagon side 2 units.
a² = 2-2cos120
a = √(3) units.
a = 1.73205080757 units.
a is the side length of the inscribed square.
Therefore;
Area Square is;
a²
= √(3)²
= 3 square units.
Calculating Area Purple.
sin30 = b/2
b = 1 unit.
c = 2b+2
c = 2*1+2
c = 4 units.
sin30 = d/1
d = ½ units.
e = 4-½-√(3)
e = ½(7-2√(3)) units.
e = 1.76794919243 units.
tan18 = f/1.76794919243
f = 0.57444151456 units.
g = 2f
g = 1.14888302912 units.
g is the side length of the inscribed regular pentagon.
h² = 2*1.14888302912²-2* 1.14888302912*1.14888302912cos108
h = 1.85893179021 units.
j = ½(1.73205080757-1.14888302912)
j = 0.29158388922 units.
k² = 0.29158388922²+1.14888302912²-2*0.29158388922*1.14888302912cos72
k = 1.09449285048 units.
(1.09449285048/sin72) = (1.14888302912/sinl)
l = 86.67708120396°
m² = 1+3-2√(3)cos60
m² = 1.50597117915 unit.
(1.50597117915/sin60) = (1/sinn)
n = 35.10390936102°
o = 360-35.10390936102-90-86.67708120396
o = 148.21900943502°
Therefore, Area Purple is;
(0.5*1.73205080757*1sin60)+(0.5*1.14888302912*0.29158388922sin72)+(0.5*1.09449285048*1.50597117915sin148.21900943502)
= 1.34335150045 square units.
It implies;
Area Purple ÷ Area Square to 2 decimal places is;
1.34335150045÷3
= 0.44778383348
≈ 0.45
