Let a be the radius of the red inscribed circle.
b = (4-a) units.
tan30 = a/b
⅓√(3) = a/(4-a)
3a = 4√(3)-√(3)a
(3+√(3))a = 4√(3)
a = 4√(3)/(3+√(3))
a = ⅙(12√(3)-12)
a = (2√(3)-2) units.
a = 1.46410161514 units.
Therefore, area red inscribed circle is;
πa²
= π(2√(3)-2)²
= π(12-8√(3)+4)
= (16-8√(3))π
= 8(2-√(3))π square units.
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