# Area of an equilateral triangle with the radii of the inscribed circle

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!

• en.wikipedia.org/wiki/… Commented Jan 31, 2021 at 16:19
• In an equilateral $\triangle$, centroid, orthocenter, incenter are the same point. Using this, $p$ and $r$ can be easily related. Or, what are $p$ and $S$ in terms of side of equilateral $\triangle$ ? Commented Jan 31, 2021 at 16:22

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 $$r$$.