Mathematics Question and Solution
Calculating inscribed regular hexagon shaded area ÷ Ascribed regular hexagon area.
Let the side length of the inscribed shaded regular hexagon be 1 unit.
a² = 2-2cos120
a = √(2) units.
Shaded inscribed area is
2(½*1*1sin120)+(1*√(3))
= ½√(3)+√(3)
= ½(3√(3)) square units.
= 2.59807621135 square units.
Let x be the side length of the ascribed regular hexagon.
b = ½(x) units.
Calculating x.
2² = x²+(½(x))²-2*x*½(x)cos120
4 = x²+¼(x²)+½(x²)
16 = 4x²+x²+2x²
7x² = 16
x = ⅐(4√(7)) units.
x = 1.51185789204 units.
sin60 = c/1.51185789204
c = 1.30930734142 units.
cos60 = d/1.51185789204
d = 0.75592894602 units.
Area ascribed regular hexagon is;
2(0.5(2*0.75592894602+1.51185789204+1.51185789204)*1.30930734142)
= 2(0.5*4.53557367612*1.30930734142)
= 2(2.96922995585)
= 593845991170 square units.
Therefore, shaded fraction is;
Inscribed regular hexagon shaded area ÷ Ascribed regular hexagon area.
= 2.59807621135÷5.9384599117
= 0.4375
= 7/16.
