Mathematics Question and Solution
Let the side length of the regular pentagon be 1 unit.
z = √(2-2cos108)
z = 1.61803398875 units.
sin18 = y/1
y = 0.30901699437 units.
cos18 = x/1
x = 0.9510565163 units.
Area Regular Pentagon is;
0.5sin108+0.5*(1+1.61803398875)*0.9510565163
= 0.47552825815+1.24494914245
= 1.7204774006 square units.
(1/sin54) = (a/sin18)
a = 0.38196601125 units.
b =1-0.38196601125
b = 0.61803398875 units.
(1/sin54) = (c/sin108)
c = 1.17557050458 units.
sin36 = 0.61803398875/d
d = 1.05146222424 units.
e = 1.05146222424+1.17557050458
e = 2.22703272882 units.
tanf = 2.22703272882/1
f = 65.81858573659°
g = 90-65.81858573659
g = 24.18141426341°
Calculating alpha to 2 decimal places.
Let it be h.
h = 36-24.18141426341
h = 11.81858573659°
h ≈ 11.82°
tan36 = 0.61803398875/j
j = 0.85065080835 units.
cos11.81858573659 = 0.85065080835/k
k = 0.86907431265 units.
tan11.81858573659 = l/0.85065080835
l = 0.17799821112 units.
m = 0.61803398875-0.17799821112
m = 0.44003577763 units.
It implies;
Area Burgundy is;
0.5*1*2.22703272882-0.5*0.44003577763*1.05146222424sin54
= 0.92635796944 square units.
Therefore;
Area Burgundy ÷ Area Regular Pentagon to 2 decimal places is;
0.92635796944÷1.7204774006
= 0.53843076876
≈ 0.54
