Mathematics Question and Solution
Let the regular pentagon side length be 1 unit.
a =⅕(180(5-2))
a = ⅕(180*3)
a = 108°
a is an angle of the regular pentagon.
Calculate x.
b = 108-90
b = 18°
x = ½(180-18)
x = ½(162)
x = 81°
c = 108-60
c = 48°
d = 180-18-48
d = 114°
(e/sin48) = (1/sin114)
e = 0.8134732862 units.
f = 1-e
f = 1-0.8134732862
f = 0.1865267138 units.
(0.8134732862/sin48) = (g/sin18)
g = 0.3382612127 units.
sin30 = 0.25/h
h = 0.5 units.
j = 0.5-g
j = 0.5-0.3382612127
j = 0.1617387873 units.
Calculating Area Purple.
k = 180-48-60
k = 72°
(0.5/sin72) = (l/sin60)
l = 0.4552964987 units.
m = 1-l
m = 1-0.4552964987
m = 0.5447035013 units.
(n/sin18) = (1/sin81)
n = 0.3128689301 units.
o = 108-81
o = 27°
p² = 0.3128689301²+0.5447035013-2*0.3128689301*0.5447035013cos27
p = 0.3014909276 units.
(0.3014909276/sin27) = (0.5447035013/sinq)
q = 55.1072632083°
r = 180-q
r = 180-55.1072632083
r = 124.8927367917°
s = 180-27-124.8927367917
s = 28.1072632083°
t = 180-28.1072632083-72
t = 79.8927367917°
u = 270-81-124.8927367917
u = 64.1072632083°
v = 180-64.1072632083-79.8927367917
v = 36°
(0.3014909276/sin36) = (w/sin79.8927367917)
w = 0.5049668583 units.
(0.5049668583/sin79.8927367917) = (y/sin64.1072632083)
y = 0.4614358732 units.
(0.5/sin72) = (z/sin48)
z = 0.3906943556 units.
A = y-z
A = 0.4614358732-0.3906943556
A = 0.0707415176 units.
B = 0.5-A
B = 0.5-0.0707415176
B = 0.4292584824 units.
Therefore;
Area Purple is;
0.5*0.4292584824*0.1617387873sin120
= 0.030063096 square units.
Calculating Area Red.
tan30 = 0.25/C
C = 0.4330127019 units.
Therefore;
Area Red is;
2(0.5*0.25*0.4330127019) + (0.5*0.4330127019)
= 0.3247595264 square units.
It implies;
Area Red ÷ Area Purple to 1 decimal place is;
0.3247595264÷0.030063096
= 10.8025975244
≈ 10.8
Therefore, Answers;
Angle x exactly is;
= 81°
Area Red ÷ Area Purple to 1 decimal place is;
= 10.8
