Mathematics Question and Solution
Let the side of the regular pentagon be 1 unit.
Calculating Area Red.
a² = 1²+1²-2*1*1cos108
a = 1.61803398875 units.
b = 108-36-60
b = 12°
(c/sin12) = (1.61803398875/sin96)
c = 0.33826121272 unit.
d = 1-c
d = 0.66173878728 unit.
e = 96-60
e = 36°
tan36 = f/0.66173878728
f = 0.48078137139 unit.
Area Red is;
½(df)
= 0.5*0.48078137139*0.66173878728
= 0.15907584083 square units.
Calculating Area Lilac.
sin36 = 0.48078137139/g
g = 0.81795412443 units.
h = 1-f
h = 0.51921862861 unit.
i = atan(1/0.51921862861)
i = 62.56081994835°
j = 180-(90-36)-i
j = 63.43918005165°
Radius, r of Lilac circle is;
(r/tan30) + (r/tan(0.5*63.43918005165) = 0.81795412443
r = 0.24416896637 unit.
Area Lilac circle is;
π(0.24416896637)²
= 0.18729699178 square units.
Therefore;
Area Red ÷ Area Lilac to 2 decimal places is;
0.15907584083 ÷ 0.18729699178
= 0.84932405651
≈ 0.85
