Mathematics Question and Solution
Let the green square side be 1 unit.
Area Green Square is;
1²
= 1 square units.
Calculating the total shaded area.
a² = 1²*2
a = √(2) units.
a is the radius of the quarter circle.
tan15 = b/1
b = (2-√(3)) units.
c = 1-b
c = (√(3)-1) units.
cos15 = 1/d
d = (√(6)-√(2)) units.
tan30 = e/√(2)
e = ⅓(√(6)) units.
Shaded Area is;
Area sector with radius √(2) units and angle 30° - Area with height (√(6)-√(2)) units and base (√(3)-1)sin105 units + Area triangle with height ⅓(√(6)) units and base sin(15) units.
= (30π*2/360)-0.5*(√(6)-√(2))(√(3)-1)sin105+0.5*⅓(√(6))sin15
= ⅙(π)-½(√(3)-1)+(3-√(3))/12
= (2π-6√(3)+6+3-√(3))/12
= (2π+9-7√(3))/12 square units
= 0.26323580452 square units.
Therefore;
Total Area Shaded ÷ Area Square as a single fraction is;
((2π+9-7√(3))/12)÷1
= (2π+9-7√(3))/12
= 0.26323580452 exactly in decimal.
= 0.26 to 2 decimal places.
