Mathematics Question and Solution
Let a be the radius of the inscribed circle.
Let the circle's centre be X.
Calculating a.
(3-a)² = √(3)²+a²
9-6a+a² = 3+a²
6 = 6a
a = 1 unit.
a is the radius of the inscribed circle (XT = XN = XE).
tanb = √(3)/1
b = atan(√(3)
b = 60°
b is angle XTH
c = 180-60
c = 120°
c is angle EXT.
It implies, area grey is;
Area sector with radius 1 unit and angle 120° - Area triangle with height 1 unit and base sin120 units.
= (120π*1*1/360)-(½*1*1sin120)
= ⅓(π)-½*½(√(3))
= ⅓(π)-¼√(3)
= (4π-3√(3))/12 square units.
= 0.6141848493 square units.
≈ 0.6142 to 4 decimal places.
