Mathematics Question and Solution
Let the a side length of the yellow regular triangle be 2 units.
Therefore, a side length of the regular pentagon is 2 units.
Area Yellow is;
0.5*2*2sin60
= 2sin60
= √(3) square units.
= 1.73205080757 square units.
Calculating r, radius of the circle.
a = 108+0.5*60
a = 108+30
a = 138°
cos30 = 1/b
b = 1.15470053838 units.
r² = 2²+1.15470053838²-2*2*1.15470053838cos138
r = 2.96070468869 units.
(2.96070468869/sin138) = (1.15470053838/sinc)
c = 15.12743321513°
d = 180-108-15.12743321513
d = 56.87256678487°
e = 90-56.87256678487
e = 33.12743321513°
f² = 1.15470053838²+2.96070468869²-2*1.15470053838*2.96070468869cos33.12743321513
f = 2.09117918124 units.
(2.09117918124/sin33.12743321513) = (1.15470053838/sing)
g = 17.56371930639°
h = 108+60
h = 168°
j = ½(180-168)
j = ½(12)
j = 6°
k = 60-6
k = 54°
l = 2r
l = 2*2.96070468869
l = 5.92140937738 units.
m² = 8-8cos168
m = 3.97808758147 units.
(5.92140937738/sinn) = (3.97808758147/sin17.56371930639)
n = 26.69115252153°
o = 180-26.69115252153
o = 153.30884747847°
p = 153.30884747847-54
p = 99.30884747847°
q = 180-99.30884747847-17.56371930639
q = 63.12743321514°
(2.09117918124/sin63.12743321514) = (s/sin99.30884747847)
s = 2.31346122073 units.
Therefore, area red is;
0.5*2.31346122073*2.09117918124sin17.56371930639
= 0.72995173554 square units.
It implies;
Area Red/Area Yellow is;
0.72995173554÷√(3)
= 0.72995173554÷1.73205080757
= 0.42143783101
≈ 0.42
