Question

A circle is inscribed in an equilateral triangle of side 8 m. The approximate area of the unoccupied space inside the triangle is

Answer 1

The center of the inscribed circle = O = incenter = centroid of the triangle ABC, as it is equilateral.   Given AB = 8m

Draw the median = perpendicular from A to BC to meet BC at D.  Draw a perpendicular from O to AB to meet AB at E.  OE is the radius of incircle.

Altitude = AD = √3/2 * AB
AO = 2/3 * AD = AB/√3
AE = AB/2
In the Right angle triangle  AOE, AO^2 = AE^2 + OE^2
OE^2 = AB^2 /3  - AB^2/4 = AB^2 /12

Area of Incircle = π OE^2 = π AB^2/12
Area of Equilateral Triangle = 1/2 * √3/2 * AB^2

Unoccupied space inside the triangle = AB^2 [ √3/4 - π/12 ]
      = 10.957 m^2

Answer 2

yes he is right......