Mathematics Question and Solution
Calculating Area Blue.
Notice.
The side lengths of the regular octagon and regular hexagon is 4 units each.
a = ⅙*180(6-2)
a = 120°
a is the single interior angle of the regular hexagon.
b = ⅛*180(8-2)
b = 135°
b is the single interior angle of the regular octagon.
c² = 4²+4²-2*4*4cos120
c² = 32+16
c² = 48
c =4√(3) units.
tand = 2/(4√(3))
d = atan(1/(2√(3))
d = 16.102113752°
e = 120-30-d
e =120-30-16.102113752
e = 73.897886248°
f = 180-e
f = 180-73.897886248
f = 106.102113752°
g = 360-a-b
g = 360-120-135
g = 105°
h = g-90
h = 105-90
h = 15°
j = 90-h
j = 90-15
j = 75°
k = f-j
k = 106.102113752-75
k = 31.102113752°
sin15 = l/4
l = 4sin15
l = 1.03527618041 units.
cos15 = m/4
m = 4cos15
m = 3.86370330516 units.
n²+n² = 4²
2n² = 16
n² = 8
n = 2√(2) units.
o = 4+n
o = (4+2√(2)) units.
o = 6.82842712475 units.
p = o+l
p = 6.82842712475+1.03527618041
p = 7.86370330516 units.
tank = q/p
tan31.102113752 = q/7.86370330516
q = 7.86370330516tan31.102113752
q = 4.74408479621 units.
Therefore, area blue is;
Area triangle with height 2√(2) units and base 2√(2) units + Area trapezoid with parallel lengths 7.86370330516 units and 6.82842712475 units, and height 3.86370330516 units + Area triangle with height 7.86370330516 and base 4.74408479621 units.
½(2√(2)*2√(2))+½((6.82842712475+7.86370330516)*3.86370330516)+½(7.86370330516*4.74408479621)
= 4+28.3830164509+18.653037646
= 51.0360540969 square units.
