an equilateral triangle has side 6 cm.find the radius of the incircle inscribed in it.

Dear Student,

We can draw the following figure for the given problem


As $△ABC$ is an equilateral triangle, all of the Altitude, Median, Angle Bisector are same.
QB is $\perp to AC and also the angle bisector of \angle ABC$
As $△ABC$ is equilateral triangle
, $$\begin{gathered} \angle ABC ={60}^{\circ } \\ \angle OBP={30}^{\circ } (QB is angle bisector) \\ in △OBP, \\ \tan 30=\frac{OP}{BP} \\ BP=3 cm, AP is median on line BC \\ OP is radius of incircle \\ \frac{1}{\sqrt{3}}=\frac{OP}{3} \\ OP=\sqrt{3} cm \end{gathered}$$
Hence Radius of in-circle is $\sqrt{3} cm$

Regards,