Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the single side length of the square be 2 unit.
Calculating area S.
It is;
Area trapezoid of two parallel side lengths 2 unit and (2-√(3)) unit and height 1 unit - area sector of radius 2 unit and angle 60° + area triangle of height √(3) unit and base 2 unit.
Notice;
√(2²-1²) = √(3) unit.
Area S is;
½(4-√(3))-(4π)/6+½(2√(3))
= 2-½(√(3))-⅔(π)+√(3)
= (12+3√(3)-4π)/6 square unit.
Calculating area R.
It is;
Area square of single side length 2 unit - area square of single side length 1 unit - area triangle of height 1unit and base 1unit - Area trapezoid of two parallel side lengths 2 unit and (2-√(3)) unit and height 1 unit - area sector of radius √(2) unit and angle 45°
= 4 - 1- ½ - ½(4-√(3)) - (2π)/8
= 1 - ½ +½√(3) - ¼(π)
= (2+2√(3)-π)/4 square unit.
Therefore;
Area R ÷ Area S in terms of π in exact fraction is;
(2+2√(3)-π)/4 ÷ (12+3√(3)-4π)/6
= 3(2+2√(3)-π)/2(12+3√(3)-4π)
= 0.75246816946 exactly in decimal.
