Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Calculating Exactly Green Area.
Let x be the side length of each of the three inscribed congruent squares.
a² = 2x²
a = √(2)x
a is the diagonal of the squares.
b = 45-30
b = 15°
cos15 = c/(√(2)x)
c = ½(√(3)+1)x units.
c = 1.36602540378x units.
d = c+x
d = ½(1+√(3))x+x
d = ½(3+√(3))x units.
d is the height of the ascribed regular hexagon.
e² = 2²+2²-2*2*2cos120
e = √(8+4)
e = 2√(3) units.
e = d.
Calculating x.
Again, e = d.
½(3+√(3))x = 2√(3)
x = 4√(3)/(3+√(3))
x = 2(√(3)-1) units.
x = 1.46410161514 units.
x' = 2-x
x' = 2-(2√(3)-2)
x' = (4-2√(3)) units.
Recall.
c = ½(√(3)+1)x units.
And x = 2(√(3)-1) units.
c = ½(√(3)+1)*2(√(3)-1)
c = (√(3)+1)(√(3)-1)
c = 3-1
c = 2 units.
tan15 = f/2
f = 2tan15
f = 2(2-√(3)) units.
f = 0.53589838486 units.
g² = (4-2√(3))²+(2√(3)-2)²
g = 1.55909609016 units.
g = 2√(11-6√(3)) units.
tanh = (2√(3)-2)/(4-2√(3))
h = atan(1+√(3))°
sin60 =j/(4-2√(3))
j = (2√(3)-3) units.
j = 0.46410161514 units.
k = x-j
k = (2√(3)-2)-(2√(3)-3)
k = 1 units.
Therefore, green area exactly in square units is;
Area triangle with height (2√(3)-2) units and base (4-2√(3))sin30 units+Area triangle with height 1 unit and base (4-2√(3)) units+Area triangle with height (2√(3)-2) units and base 2sin30.
= (½*(2√(3)-2)(4-2√(3))sin30)+(½*1*(4-2√(3)))+(½*2*(2√(3)-2)sin30)
= ((√(3)-1)(2-√(3)))+(2-√(3))+(√(3)-1)
= ((√(3)-1)(2-√(3)))+1
= 3√(3)-5+1
= (3√(3)-4) square units.
= 1.19615242271 square units in decimal.
