Mathematics Question and Solution
Let the side length of the regular hexagon be 2 units.
a = ⅙(180(6-2))
a = 120°
a is the single interior angle of the regular hexagon.
b² = 2²+1²-2*2*1cos120
b = √(7) units.
b = 2.6457513111 units.
(2.6457513111sin120) = (1/sinc)
c = 19.1066053509°
d = 120-c
d = 120-19.1066053509
d = 100.8933946491°
Therefore, e is;
e = 0.5*√(7)*2cos100.8933946491
e = √(7)cos100.8933946491
e = 2.5980762114 square units.
e is the area of green plus orange inscribed plane shape (inscribed scalene triangle).
Calculating green area.
f² = 2²+2²-2*2*2cos120
f = √(12)
f = 2√(3) units.
tang = 2√(3)
g = atan(2√(3))°
g = 73.897886248°
h = 180-120-c
h = 60-19.1066053509
h = 40.8933946491°
j = g-h
j = 73.897886248-40.8933946491°
j = 33.0044915989°
k = 40.8933946491+73.897886248
k = 114.7912808971°
l = 180-114.7912808971
l = 65.2087191029°
It implies;
(1/sin65.2087191029) = (m/sin73.897886248)
m = 1.0583005244 units.
n = 180-114.7912808971-33.0044915989
n = 32.204227504°
It implies;
(1.0583005244/sin32.204227504) = (o/sin114.7912808971)
o = 1.8027756377 units.
Therefore, green area is;
Let it be p.
p = 0.5*1.8027756377x1.0583005244sin33.0044915989
p = 0.5196152422 units.
Again, p is area green.
Therefore, area orange, q is;
q = e-p
q = 2.5980762114-0.5196152422
q = 2.0784609692 square units.
Again, q is area orange.
Therefore;
Area Orange : Area Green is;
= q : p
= 2.0784609692 : 0.5196152422 2.0784609692÷0.5196152422
= 4
It implies;
Area Orange : Area Green is 4 : 1
