Mathematics Question and Solution
Let the side length of the regular hexagon be 1 unit.
a² = 2-2cos120
a = √(3) units.
Area regular hexagon is;
2(½*sin120)+(1*√(3))
= ½√(3)+√(3)
= ½(3√(3)) square units.
= 2.5980762114 square units.
Calculating yellow area.
b² = 2(0.5)²-2(0.5)²cos120
b = ½√(3) units.
c = ½(a-b)
c = ½(√(3)-½√(3))
c = ¼√(3) units.
cos60 = d/0.5
d = ¼ units.
e = c-d
e = ¼√(3)-¼
e = ¼(√(3)-1) units.
Area yellow is;
Area regular hexagon with side length 1 unit - Area triangle with height 1 unit and base sin120 units - 2(area rectangle with length ¼√(3) units and width ¼(√(3)-1) units) - Area triangle with length ½√(3) units and width ¼(√(3)-1) units + Area square with side ¼√(3) units + Area rectangle with length ½√(3) units and width ¼√(3) units.
= ½(3√(3))-¼√(3)-2(¼√(3)*¼(√(3)-1))-(½√(3)*¼(√(3)-1)+(¼√(3)*¼√(3))+(½√(3)*¼√(3))
= ½(3√(3))-¼√(3)-2((3/16)-(√(3)/16))-((3/8)-(√(3)/8))+(3/16)+(3/8)
= ½(3√(3))-¼√(3)-(3/8)+(√(3)/8)-(3/8)+(√(3)/8)+(9/16)
= ½(3√(3))-¼√(3)+¼√(3)-(¾)+(9/16)
= ½(3√(3))-(¾)+(9/16)
= ½(3√(3))-(3/16)
= (24√(3)-3)/16 square units.
= 2.4105762114 square units.
Or
Yellow Area is;
½(3√(3))-(0.5³sin120)-2(0.5²*(¼√(3)-¼)sin120)
= 2.5980762114-0.1082531755-0.0792468245
= 2.4105762113 square units.
Therefore, the fraction of the shaded regular hexagon is;
2.4105762113÷2.5980762114
= 0.9278312163
= (24-√(3))/24
