Mathematics Question and Solution
Let the side length of the three congruent regular hexagon be 1 unit.
a = ⅙(180(6-2))
a = 120°
a is the single interior angle of the three congruent regular hexagon.
b² = 2-2cos120
b = √(3) units.
tanc = 2√(3)/1
c = atan(2√(3))
c = 73.897886248°
d = c-30
d = 73.897886248-30
d = 43.897886248°
tan43.897886248 = e/√(3)
e = ⅓(5) units.
f² = (⅓(5))²+(√(3))²
f = ⅓(2√(13)) units.
f = 2.4037008503 units.
g² = 1²+2/(2√(3))²
g = √(13) units.
g = 3.6055512755 units.
h = g-f
h = √(13)-⅓(2√(13))
h = ⅓(√(13)) units.
h = 1.2018504252 units.
i = 90-d
i = 90-43.897886248
i = 46.102113752°
j = 180+120-i
j = 60-46.102113752
j = 13.897886248°
Therefore, total red area is;
2(area triangle with height and base √(3) units and (5/3) units respectively)+2(area triangle with height 1.2018504252 units and base sin13.897886248 units)+Area equilateral triangle with height 0.5 units and base 0.5sin60 units.
= 2(½*(5/3)*√(3))+2(0.5*1*1.2018504252sin13.897886248)+(0.5*0.5*0.5sin60)
= 2.8867513459+0.2886751346+0.1082531755
= 3.283679656 square units.
Calculating the area of the complete composite plane shape.
It is;
Area regular hexagon with side 1 unit+4(area triangle with height 1 unit and base sin120 units)+2(area triangle with height √(3) units and base (5/3) units)+2(area triangle with height √(3) units and base 0.5 units)+Area triangle with height 0.5 units and base 0.5sin60 units+Area triangle with height 1 unit and base sin60 units.
2(0.5(1+2)*½√(3))+4(0.5*1*1sin120)+2(√(3)*(5/3))+2(0.5*√(3)*0.5)+(0.5*0.5*0.5sin60)+(0.5*1*1sin60)
= 2.5980762114+1.7320508076+2.8867513459+0.8660254038+0.1082531755+0.4330127019
= 8.6241696461 square units.
Therefore, shaded area fraction is;
Total red area ÷ Composite plane shape area
= 3.283679656÷8.6241696461
= 0.3807531381
≈ 0.38 to 2 decimal places.
