Mathematics Question and Solution
Notice.
The ascribed plane shape is a regular pentagon with side length 10 units.
Let a be the radius of the bigger inscribed circle.
Radius of the smaller pink inscribed circle is r.
c = ⅕(180(5-2))
c =108°
c is the single interior angle of the ascribed regular pentagon.
d² = 10²+10²-2*10*10cos108
d = 16.1803398875 units.
Calculating a, radius of the bigger inscribed circle.
tan(0.5*108) = a/(0.5*10)
a = 5tan(54)
a = 6.8819096024 units.
d² = e²+5²
16.1803398875² = e²+25
e = √(16.1803398875²-25)
e = 15.3884176859 units.
f = e-2a
f = 15.3884176859-2(6.8819096024)
f = 1.6245984811 units.
Calculating r, radius of the smaller inscribed pink circle.
sin(0.5*108) = r/(1.6245984811-r)
sin(54) = r/(1.6245984811-r)
r = 1.6245984811sin(54)-sin(54)r
r+sin(54)r = 1.6245984811sin(54)
1.8090169944r = 1.3143277802
Therefore;
r = 1.3143277802/1.8090169944
r = 0.726542528 units.
r = 0.73 units to 2 decimal places.
