Mathematics Question and Solution
Let the single side length of the two congruent inscribed regular pentagon be 1 unit.
Area Green is;
0.5*5(1/(2tan(180/5)))
= 1.72047740059 square units.
Calculating Area Shaded.
a = ⅕(180*3)
= 108°
Where a is the single interior angle of the congruent inscribed regular pentagon.
b² = 2-2cos108
b = 1.61803398875 units.
c = 1+b
c = 2.61803398875 units.
d² = 2.61803398875²+1²-2*1*2.61803398875cos72
d = 2.49721204096 units.
Where d is the radius of the inscribed arc.
2.49721204096² = 1.61803398875²+e²-2e*1.61803398875cos144
e²+2.61803398875-3.61803398877 = 0
e = 1 unit.
f =½(360-2(108))
f = 72°
g² = 2-2cos72
g = 1.17557050458 units.
h = asin(0.5*1.17557050458/2.49721204096)
h = 13.61382244073°
i = 2h
i = 27.22764488145°
Where i is the angle of the inscribed arc.
Area Shaded is,
2(area triangle with height 1 unit and base sin72 units) + Area sector with radius 2.49721204096 units and angle 27.22764488145° - Area triangle with height 2.49721204096 units and base 2.49721204096sin27.22764488145 units.
= 2(0.5sin72)+27.22764488145π*2.49721204096*2.49721204096/360-0.5*2.49721204096*2.49721204096sin27.22764488145
= 1.00619906795 square units.
It implies;
Area Shaded ÷ Area Green to 2 decimal places is;
1.00619906795÷1.72047740059
= 0.5848371316
≈ 0.58
