Mathematics Question and Solution
Calculating Area Blue, Inscribed Square Area.
Notice.
2 units is the radius of the inscribed circle.
tan60 = 2/a
√(3) = 2/a
a = ⅓(2√(3)) units.
a = 1.15470053838 units.
sin60 = 2/b
½√(3) = 2/b
4 = √(3)b
b = ⅓(4√(3)) units.
b = 2.30940107676 units.
c = a+b
c = ⅓(2√(3))+⅓(4√(3))
c = ⅓(6√(3))
c = 2√(3) units.
Let r be the radius of the ascribed quarter circle.
d = ½(r) units.
e = c+d
e = (2√(3)+½(r)) units.
f = (r-2) units.
Calculating r.
(r-2)² = 2²+(2√(3)+½(r))²
r²-4r+4 = 4+12+2√(3)r+¼(r²)
r²-4r = 12+2√(3)r+¼(r²)
¼(3r²)-(2√(3)+4)r-12 = 0
3r²-(8√(3)+16)r-48 = 0
3r²-29.8564064606r-48 = 0
It implies;
r = 11.361 units.
Again, r is the radius of the ascribed quarter circle.
d = ½(r) units.
And r = 11.361 units.
d = 0.5(11.361)
d = 5.6805 units.
d is half the radius of the ascribed quarter circle.
g² = 2²+a²
g² = 2²+(⅓(2√(3)))²
g² = 4+⅑(12)
g² = 4+(4/3)
g = √(16/3)
g = ⅓(4√(3)) units.
g = b.
h² = d²+g²-2dgcos(180-60)
h² = 5.6805²+2.30940107676²-2*5.6805*2.30940107676cos120
h = 7.12179516694 units.
It implies, x², blue area (area inscribed square) with x it side length is;
h² = 2²+x²
x² = 7.12179516694²-2²
x² = 46.7199663998 square units.
Where x, the side length of the blue inscribed square area is;
x = √(46.7199663998)
x = 6.83520053837 units.
