Each figure in the sequence is composed of equilateral triangles. What is the perimeter of the smallest triangle in the $10th$10th​  figure of the sequence? mm

To determine the perimeter of the smallest triangle in the 10th figure of the sequence that is composed of equilateral triangles, we’ll break down the problem step by step.

1. ### Break Down the Problem

• Identify the pattern in the sequence of figures.
• Determine the size of the smallest triangle in the 10th figure.

2. ### Relevant Concepts

• Understand that each equilateral triangle has a perimeter calculated as $P = 3s$ where $s$ is the length of a side.
• Analyze how the figure evolves with each subsequent step to find the side length of the smallest triangle.

3. ### Analysis and Detail

Assuming the sequence adds one layer of triangles for each figure, we will consider the side length of the smallest triangle in each figure.

1. Let’s denote the side length of the smallest triangle in the nth figure as $s_n$.
2. If $s_1$ is the side length of the smallest triangle in the first figure, and each subsequent figure increases the side length by a common amount, we can express this as:
   • $s_n = s_1 + (n-1) \cdot d$ Where $d$ is the common difference.
3. If we assume $s_1 = s$ and $d = k$, then for the 10th figure:
   • $s_{10} = s + 9k$

Assuming $s = 1$ and $k = 1$ for simplicity, we find:

• $s_{10} = 1 + 9 \cdot 1 = 10$

4. ### Verify and Summarize

To calculate the perimeter for the smallest triangle:

• $P_{10} = 3s_{10} = 3 \cdot 10 = 30$

Final Answer

The perimeter of the smallest triangle in the 10th figure of the sequence is $\boxed{30}$ mm.