Find the area of an equilateral triangle $ABC$ with the radii of the inscribed circle $r$.
The answer given in my book is $3r^2\sqrt3$.
We know that $S=pr$ where $p$ is the semiperimeter and $r$ is the radii of the inscribed circle. How can I express $p$ with $r$? Thank you in advance!
Welcome to MSE. Since $p = 3a/2$, where $a$ is the length of a side of the triangle, so it suffices to relate $r$ to $a$.
To do this, see the figure here below equation (2). We can make a right triangle with legs of length $a/2$ and $r$, with one angle equal to $30°$. You can now use trigonometry to solve for $a$ in terms of $r$.