Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the single side length of the large square be 2 units.
Therefore;
Area large square is;
2 x 2 = 4 square units.
Calculating Area R.
It will be;
Area trapezoid of two parallel lengths (√(2)-1+2) units and height (3-√(2)) units + Area rectangle of length 2 units and width (√(2)-1) units - 2( area triangle of height (√(2)-1) units and base 1 unit) - Area sector of radius 1 unit and angle 45° - Area square of single side length 1 unit - Area trapezoid of two parallel lengths (√(2)-1+½(√(2)-1)) units and height 1 unit.
= ½(√(2)+1)(3-√(2)) + 2(√(2)-1) - (1 x 1) - (45π/360) - ½(√(2)-1+½(√(2)-1)) x 1
= ½(2√(2)+1) + 2√(2) - 2 - (√(2)-1) - 1 - (π/8) - ¼(3√(2)-3)
= (8√(2)+4+16√(2)-16-8√(2)+8-8-π-6√(2)+6)/8
= (10√(2) - 6 - π)/8
Area R is;
(10√(2) - 6 - π)/8 square units.
Area R ÷ Area large square is;
((10√(2) - 6 - π)/8)÷4
= (10√(2)-6-π)/32
