Mathematics Question and Solution
Diameter of the inscribed circle is;
½(AB)
= ½(8)
= 4 units.
a = ½(4)
a = 2 units.
a is the radius of the inscribed circle.
Calculating red length.
Notice, the red length is the side of the inscribed regular hexagon.
Let the red length be b.
c² = 2b²-2b²cos120
120° is the single interior angle of the inscribed regular hexagon.
c = √(3)b units.
d = ½(b) units.
e² = 2(2)²
e = √(8)
e = 2√(2) units.
f² = 2(8)²
f = 8√(2) units.
f is the diagonal of the ascribed square.
Therefore, red length is;
c+d+e+2 = f
√(3)b+½(b)+2√(2)+2 = 8√(2)
(√(3)+½)b = 6√(2)-2
½(2√(3)+1)b = 6√(2)-2
(2√(3)+1)b = (12√(2)-4)
b = (12√(2)-4)/(2√(3)+1)
b = (24√(6)-8√(3)-12√(2)+4)/(12-1)
b = (24√(6)-8√(3)-12√(2)+4)/11 units.
b = 2.9055258743 units.
Again, b is the red length, also, it is the side length of the inscribed regular hexagon.
2g² = b²
g² = ½(2.9055258743)²
g = 2.0545170486 units.
h = 8-g
h = 5.9454829514 units.
j = 180-45-120
j = 15°
tan15 = k/h
k = 5.9454829514tan15
k = 1.5930873554 units.
Area Purple is;
½*1.5930873554*5.9454829514
= 4.7358368559 square units.
Calculating Area Orange.
tan(0.5*45) = 2/l
l = 4.8284271247 units.
m = l-0.5b
m = 4.8284271247-0.5*2.9055258743
m = 3.3756641876 units.
(2.9055258743/sin105) = (n/sin15)
n = 0.7785333116 units.
(2.9055258743/sin105) = (o/sin60)
o = 2.6050232326 units.
p = 2.9055258743+0.7785333116
p = 3.6840591859 units.
q = 8-2.6050232326-2.0545170486
q = 3.3404597188 units.
r = 8-4.8284271247-2
r = 1.1715728753 units.
Area Orange is;
Area triangle with height 1.1715728753 units and base 3.3404597188 units + Area triangle with height 3.3756641876 units and base 3.6840591859sin60 units.
=½(1.1715728753*3.3404597188)+½(3.3756641876*3.6840591859sin60)
= 1.9567959988+5.3850094659
= 7.3418054647 square units.
Area Orange ÷ Area Purple to 2 decimal places is;
7.3418054647÷4.7358368559
= 1.5502657055
≈ 1.55
It implies;
Answers.
Red length exactly = (24√(6)-8√(3)-12√(2)+4)/11 units.
= 2.9055258743 units exactly in decimal.
Area Orange ÷ Area Purple to 2 decimal places = 1.55
