Find the side length of an equilateral triangle whose incircle has the radius 3𝑚3​ m.

1. Break Down the Problem

To find the side length $s$ of an equilateral triangle with a given incircle radius $r$, we need to understand the relationship between the radius of the incircle and the side length.

2. Relevant Concepts

For an equilateral triangle, the relationship between the side length $s$ and the radius $r$ of the incircle is given by the formula: $r = \frac{s \sqrt{3}}{6}$ Here, $r$ is the radius of the incircle.

3. Analysis and Detail

We are provided with the radius of the incircle: $r = 3 \, m$

We can rearrange the formula to solve for the side length $s$: $s = \frac{6r}{\sqrt{3}}$

Now plug in $r = 3$: $s = \frac{6 \times 3}{\sqrt{3}} = \frac{18}{\sqrt{3}} = 6\sqrt{3} \, m$

4. Verify and Summarize

Ensure that the calculations are correct:

1. Substitute $r = 3$ into the rearranged formula.
2. Calculate: $s = \frac{18}{\sqrt{3}} = 6\sqrt{3} \, m \approx 10.39 \, m$

Final Answer

The side length of the equilateral triangle is $6\sqrt{3} \, m$ or approximately $10.39 \, m$.