Mathematics Question and Solution
Notice!
The ascribed polygon is a regular dodecagon (12 sides and 12 interior angles equal).
It's since interior angle is;
a = 180(12-2)/12
a = 1800/12
a = 150°
The middle inscribed polygon is a regular octagon (8 sides and 8 interior angles equal).
It's since interior angle is;
b = ⅛*180(8-2)
b = ⅛*(1080)
b = 135°
The completely inscribed polygon is a regular hexagon (6 sides and 6 interior angles equal).
It's since interior angle is;
c = ⅙*180(6-2)
c = ⅙*(720)
c = 120°
Notice again!
The three regular polygons has equal sides, let it be d.
Calculating d.
Considering the regular octagon.
d² = 2e²
e² = d²/2
e = ½√(2)d unit.
f = 2e+d
f = 2(½√(2)d)+d
f = (d+√(2)d) units.
g = 7+f
g = (7+d+√(2)d) units.
h = ½(a)
h = ½(150)
h = 75°
It implies;
tanh = g/d
tan75 = (7+d+√(2)d)/d
dtan(75) = 7+d+√(2)d
d(tan75+1+√(2)) = 7
d = 7/(tan75+1+√(2))
d = 5.3117333157 units.
Again, d is the side length of each of the three polygons.
f = (d+√(2)d) units.
And d = 5.3117333157 units.
f = 5.3117333157+√(2)*5.3117333157
f = 12.8236586105 units.
e = ½√(2)d unit.
And d = 5.3117333157 units.
e = 0.5√(2)*5.3117333157
e = 3.7559626474 units.
j² = 2(5.3117333157)²-2(5.3117333157)²cos120
j = 9.200191979 units.
Therefore, Painted Area Red is;
Area regular octagon with side length 5.3117333157 units - Area regular hexagon with side length 5.3117333157 units.
= ((2*0.5(5.3117333157+12.8236586105)*3.7559626474)+(5.3117333157*12.8236586105))-((2*0.5*5.3117333157²sin120)+(5.3117333157*9.200191979))
= 62.9282599727 square units.
