Mathematics Question and Solution
Let the side length of the congruent yellow regular hexagon be 1 unit.
a = 3sin30+1
a = 2.5 units.
sin60 = b/1
b = 0.5√(3) units.
c² = a²+b²
c² = 2.5²+(0.5√(3))²
c = √(7) units.
c = 2.64575131106 units.
c is the radius of the circle.
d² = √(7)²-1
d = √(6) units.
e = d-b
e = √(6)-0.5√(3)
e = 1.583464339 units.
e is the height of the green inscribed triangle.
Area Green Triangle is;
Half the height, 1.583464339 units multiply the base, 1 unit of the triangle
= 0.5*1.583464339*1
= 0.7917321695 square units.
Calculating Shaded Area.
tan(f) = (3.5)/(0.5√(3))
f = 76.10211375199°
g = 90-76.10211375199
g = 13.89788624801°
h² = (0.5√(3))²+3.5²
h = √(13) units.
h = 3.60555127546 units.
(√(7)/sin13.89788624801) = (1/sinj)
j = 5.20871910285°
cosk = √(7))/√(13)
k = 42.79413710711°
l = 42.79413710711-5.20871910285
l = 37.58541800426°
m² = 13-7
m = √(6) units.
Shaded Area is;
0.5*√(7)*√(6)-0.5√(13)sin13.89788624801-(37.58541800426π*7/360)
= 3.2403703492-0.43301270189-2.29596253219
= 0.51139511512 square units.
It implies;
Area Shaded ÷ Area Green to 2 decimal places is;
0.51139511512÷0.7917321695
= 0.64591933336
≈ 0.65
