Mathematics Question and Solution
Calculating Area Blue.
a = ⅛*180(8-2)
a = 45*3
a = 135°
a is the single interior angle of the ascribed regular octagon.
b = ⅙*180(6-2)
b = 30*4
b = 120°
b is the single interior angle of the inscribed regular hexagon.
c = a-b
c = 135-120
c = 15°
d² = 4²+4²-2*4*4cos15
d = 1.04420953776 units.
e = ½(180-c)
e = ½(180-15)
e = ½(165)
e = 82.5°
f = a-e
f = 135-82.5
f = 52.5°
g² = 1.04420953776²+4²-2*1.04420953776*4cos52.5
g = 3.46482000046 units.
(3.46482000046/sin52.5) = (4/sinh)
h = 66.3332346534°
h' = 180-h
h' = 180-66.3332346534
h' = 113.666765347°
j = 360-b-e-h'
j = 360-120-82.5-113.666765347
j = 43.833234653°
k²+k² = 4²
2k² = 16
k² = 8
k = 2√(2) units.
l = 4+2k
l = 4+4√(2)
l = 9.65685424949 units.
l is the height of the regular octagon.
m² = 4²+4²-2*4*4cos120
m = √(32+16)
m = √(48)
m = 4√(3) units.
m = 6.92820323028 units.
m is the height of the regular hexagon.
n = l-m
n = 9.65685424949-6.92820323028
n = 2.72865101921 units.
Therefore, area blue is;
Area triangle with height 4 units and base 2.72865101921 units + Area triangle with height 4 units and base 2.72865101922sin45 units + Area triangle with height 4 units and base 3.46482000046sin43.833234653 units.
= 0.5(4*2.72865101921)+0.5(4*2.72865101922sin45)+0.5(4*3.46482000046sin43.833234653)
= 5.45730203842+3.85889527836+4.79920338001
= 14.1154006968 square units.
