Mathematics Question and Solution
Let the inscribed regular pentagon side be 1 unit.
Area small/inscribed pentagon is;
0.5*5(1/(2tan(180/5)))
= 1.72047740059 square units.
Calculating Shaded Area.
a² = 2-2cos(60+108)
a = 1.98904379074 units.
b = ½(180-108-60)
b = 6°
c = 60-2(6)
c = 48°
sin60 = d/1
d = 0.86602540378 unit.
tan24 = e/0.86602540378
= 0.38557935197 unit.
f = 2e
f = 0.77115870394 units.
g = 0.5(1-f)
g = 0.11442064803 unit.
cos24 = 0.86602540378/h
h = 0.94798282509 unit.
Area Shaded is;
48*π*1.98904379074*1.98904379074/360-0.5*0.94798282509*0.94798282509sin48+2*0.5*1*0.11442064803sin108
= 1.43210804765 square units.
Area Shaded ÷ Area Small Pentagon to 2 decimal places is;
1.43210804765÷1.72047740059
= 0.83238992105
≈ 0.83
