Mathematics Question and Solution
Calculating shaded yellow area.
Let x be the side length of the ascribed regular decagon.
Let y be the side length of the two equal inscribed regular pentagons.
a = ⅒*180(10-2)
a = 144°
a is the single interior angle of the ascribed regular decagon.
b = ⅕*180(5-2)
b = 108°
b is the single interior angle of the both equal inscribed regular pentagons.
c = (1+y) units.
It implies;
(1+y)² = 2x²-2x²cos144
y = 1.90211303259x-1 --- (1).
d² = 2y²-2y²cos108
d = 1.61803398875y units.
e = 108-½(180-108)
e = 108-36
e = 72°
sin72 = f/1.61803398875y
f = 1.53884176854y units.
g = 2f
g = 3.07768353708y units.
h = a-½(5(b)-3(a))
h = 144-½(540-432)
h = 90°
j = ½(a)
j = 72°
cos72 = x/k
k = 3.2360679775x units.
Notice, it implies;
g = k
3.07768353708y = 3.2360679775x --- (2).
Calculating x.
Substituting (1) in (2).
3.07768353708(1.90211303259x-1) = 3.2360679775x
5.85410196607x-3.07768353708 = 3.2360679775x
2.61803398857x = 3.07768353708
x = 3.07768353708/2.61803398857
x = 1.17557050463 units.
Again, x is the side length of the ascribed regular decagon.
Recall.
At (1).
y = 1.90211303259x-1
And x = 1.17557050463 units.
y = (1.90211303259*1.17557050463)-1
y = 1.23606797759 units.
Again, y is the side length of each of the two equal inscribed regular pentagons.
Recall.
c = (1+y) units.
And y = 1.23606797759 units.
c = 1+1.23606797759
c = 2.23606797759 units.
l = ½(360-2(144))
l = 36°
sin36 = m/1.17557050463
m = 0.69098300565 units.
cos36 = n/1.17557050463
n = 0.95105651633 unit.
o = 2n+x
o = 2(0.95105651633)+1.17557050463
o = 3.07768353729 units.
Area ascribed regular decagon is;
2(0.5*x²sin144)+2(0.5(x+o)*m)+(o*c)
= 2(0.5*1.17557050463*1.17557050463sin144)+2(0.5(1.17557050463+3.07768353729)*0.69098300565)+(3.07768353729*2.23606797759)
= 0.81229924064+2.93892626168+6.88190960289
= 10.6331351052 square units.
Calculating the both inscribed equal regular pentagons.
Recall.
f = 1.53884176854y units.
And y = 1.23606797759 units.
f = 1.53884176854*1.23606797759
f = 1.90211303267 units.
f is the height of one of the both equal inscribed regular pentagons.
Area of the both equal inscribed regular pentagons is;
2(2(0.5*y²sin108)+(0.5*y*f))
= 2(2(0.5*1.23606797759*1.23606797759sin108)+(0.5*1.23606797759*1.90211303267))
= 2(1.45308505622+1.17557050472)
= 5.25731112188 square units.
It implies, shaded yellow area is;
Area Ascribed Regular Decagon - Area Inscribed Two Equal Regular Pentagons.
= 10.6331351052-5.25731112188
= 5.37582398332 square units.
