Question

a circle is inscribed in an equilateral triangle of 8 m.find the approx area of unoccupied space inside the triangle.

Answer 1

In an equilateral triangle ABC the centroid G of the triangle is the center of the incircle (inscribed circle).  Let the altitude be AGD, as it passes through G.
 
Let AB = BC = CA = a = 8 m.  
Altitude AD = √3 a /2 = 4√3 m

Radius of Incircle = DG = AD / 3 = 4 / √3  m

Area of incircle = 16 π / 3  m²
Area of Equilateral triangle = 1/2 * AD * BC = √3/4  * 8² = 16√3 m²

Unoccupied space in ΔABC: 16√3 - 16 π /3  = 16/3 *(3√3 - π) m²