Let r be the inradius of the triangle,
given that the area of the incircle =163pi,
=> pir^2=163pi
=> r=sqrt163
Formula for inradius (r) of a triangle : r=(2A)/p,
where A and p are the area and the perimeter of the triangle, respectively.
Recall that area A of an equilateral triangle is : A=(sqrt3a^2)/4,
where a is the side length of the equilateral triangle.
=> r=(2A)/p=((2*sqrt3a^2)/4)/(3a)=(sqrt3a)/6
=> (sqrt3a)/6=sqrt163
=> a=(6sqrt163)/sqrt3=2sqrt489 units
Footnotes: formula for inradius r of an equilateral traingle is given by :
r=sqrt3/6a,
where a is the side length of the equilateral triangle.
