Author: Sir Mike Ambrose
Let the side length of the square be 1 unit.
Area of the square = 1² = 1 square unit.
Area of Region R = Area of square − Area of equilateral triangle (side 1) − 2 × Area of a sector (radius 1, angle 30°)
= 1 − ½(sin 60°) − 2(30π/360)
Since sin 60° = √3/2,
Area R = 1 − √3/4 − π/6
= (12 − 3√3 − 2π) / 12 square units.
Therefore, the ratio of Area R to the Area of the square is:
(12 − 3√3 − 2π) / 12
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