Mathematics Question and Solution
Notice;
The inscribed star is formed from a pentagram (a regular pentagon is ascribing the star).
Let the side of the pentagram be 1 unit.
tan54 = a/0.5
a = 0.68819096024 unit.
cos54 = 0.5/b
b = 0.85065080835 unit.
c = b+a
c = 1.53884176859 units.
Let the side length of the ascribed regular hexagon be d.
Calculating d.
sin45 = e/d
e =0.5√(2)d unit.
It implies;
e+d = c
0.5√(2)d+d = 1.53884176859
d = 0.90143263769 unit.
f = 0.90143263769+2(0.90143263769sin45)
f = 2.17625089948 units.
g = ½(2.17625089948-1)
g = 0.58812544974 unit.
h² = 0.58812544974²+0.90143263769²-2*0.90143263769*0.58812544974cos45
h = 0.63931156831 unit.
(0.63931156831/sin45) = (0.90143263769/sini)
i = 85.57875793621°
j = 180-85.57875793621
j = 94.42124206379°
k = 108-94.42124206379
k = 13.57875793621°
l = 180-45-94.42124206379
l = 40.57875793621°
m = 135-40.57875793621
m = 94.42124206379°
(0.63931156831/sin72) = (n/sin94.42124206379)
n = 0.67021162252 unit.
Area Yellow is;
0.5*0.67021162252*0.63931156831sin13.57875793621+0.5*0.58812544974*0.90143263769sin45
= 0.2377372084 square units.
Calculating Shaded Area.
o = atan(0.5/1.53884176859)
o = 18°
tan36 = p/0.5
p = 0.363271264 unit.
tan54 = q/0.5
q = 0.68819096024 unit.
r = q-p
r = 0.32491969624 unit.
Where r is the radius of the inscribed circle.
Shaded Area is;
(0.5*5*1²)*(1/(2tan36)) - 5(0.5*0.363271264) - π(0.32491969624)²
= 1.72047740059-0.90817816-0.33166676119
= 0.4806324794 square units.
Area Yellow ÷ Area Shaded to 2 decimal places is;
0.2377372084÷0.4806324794
= 0.4946340886
≈ 0.49
