Mathematics Question and Solution
a² = 2²+2²-2*2*2cos120
a = √(12)
a = 2√(3) units.
a is the side length of each of the congruent 4 inscribed regular hexagon.
Calculating the side length of the ascribed regular hexagon.
sin30 = c/a
½ = c/(2√(3))
c = √(2) units.
b = a+2(2c+a)
b = 2√(3)+2(2√(3)+2√(3))
b = 2√(3)+2(4√(3))
b = 10√(3) units.
Again, b is the side length of the ascribed regular hexagon.
(10√(3))² = 2d²-2d²cos120
300 = 3d²
d² = 100
d = 10 units.
d is the side length of the ascribed regular hexagon.
sin60 = e/(2√(3))
√(3)/2 = e/(2√(3))
e = 3 units.
f = 2e
f = 2*3
f = 6 units.
Area Red is;
2(area triangle with height 10 units and base 10sin120 units) + Area rectangle with length and width 10√(3) units and 10 units respectively - 8(area triangle with height 2√(3) units and base 2√(3)sin120 units) - 4(Area rectangle with length and width 2√(3) units and 6 units respectively)
= (2*½*10*10sin60)+(10*10√(3))-8(½*2√(3)*2√(3)sin60)-4(2√(3)*6)
= 50√(3)+100√(3)-24√(3)-48√(3)
= 150√(3)-72√(3)
= 78√(3) square units.
