Mathematics Question and Solution
Let the ascribed regular pentagon side length be 1 unit.
a = ⅕(180(5-2))
a = 108°
a is the single interior angle of the ascribed and the smaller inscribed regular pentagon.
b = ½(180-108)
b = 36°
c = 108-36
c = 72°
d = 180-108
d = c = 72°
e = 180-72-72
e = b = 36°
Let f be the side length of the smaller inscribed regular pentagon.
g = (1-f) units.
Calculating f.
cos72 = (0.5(1-f))/f
0.6180339887f = 1-f
1.6180339887f = 1
f = 1/1.6180339887
f = 0.6180339887 units.
Again, f is the side length of the smaller inscribed regular pentagon.
g = (1-f)
g = 1-0.6180339887
g = 0.3819660113 units.
Observing Cosine Rule.
h² = 1²+0.3819660113²-2*0.3819660113*1cos108
h = 1.1755705046 units.
Therefore, the required angle is;
Let it be j.
Observing Sine Rule.
(1.1755705046/sin108) = (1/sinj)
j = asin(sin108÷1.1755705046)
j = 54°
