Calculating the total shaded area with respect to d.
a = (180(12-2))/12
a = 150°
a is the single interior angle of the regular dodecagon.
b = 105-75
b = 30°
sin30 = c/d
c = ½(d) units.
sin60 = e/d
e = ½√(3)d units.
f = e-c
f = ½√(3)d-½(d)
f = ½(√(3)-1)d units.
g = ½(f)
g = ¼(√(3)-1)d units.
It implies, total shaded green area in terms of d is;
2(area trapezoid with parallel side lengths ½(d) units and ½√(3)d units respectively and height ¼(√(3)-1)d units.
= 2(½(½(d)+½√(3)d)*¼(√(3)-1)d)
= 2(¼(√(3)+1)d*¼(√(3)-1)d)
= (2(2d²)/16)
= 2(⅛(d²))
= ¼(d²) square units.
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