Mathematics Question and Solution
Let the side length of the square be 2 unit.
Therefore area ABD (equilateral triangle) is;
Area triangle with height 2 unit and base (2sin60) unit.
= ½*2*2sin60
= √(3) square unit.
Area purple is;
Area triangle with height 2√(5-2√(3)) unit and base (√(3)/3sin(atan(⅓(2+√(3)))))sin(atan(⅓(2+√(3)))+atan(½(√(3)+1))).
= ½(2√(5-2√(3)))(√(3)/3sin(atan(⅓(2+√(3)))))sin(atan(⅓(2+√(3)))+atan(½(√(3)+1)))
= ½*(2√(5-2√(3)))(√(3)/3sin(atan(⅓(2+√(3)))))sin(105)
= (√(5-2√(3))*√(3)*sin105)/(3sin(atan(⅓(2+√(3))))) square unit.
Therefore;
Area Purple ÷ Area ABD is;
(√(5-2√(3))*√(3)*sin105)/(3sin(atan(⅓(2+√(3))))) ÷ √(3)
= (√(5-2√(3))sin105)/(3sin(atan(⅓(2+√(3)))))
= 0.51196612829
≈ 0.512 to 3 decimal places.
