In the following figure, the perimeter of the larger equilateral triangle is $24 \mathrm{~cm}$ and each of the smaller triangles are also equilateral. What is the sum of the lengths of all the segments drawn in the figure?

![](https://cdn.mathpix.com/cropped/2024_05_01_3fc4f3a6ef7302a792fcg-25.jpg?height=450&width=516&top_left_y=1792&top_left_x=839)

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Solution

Since the perimeter of the larger triangle is $24 \mathrm{~cm}$, it follows that the side of each of the smaller triangles is $1 \mathrm{~cm}$. To find the sum of all segments, simply add the perimeters of the gray triangles and the black triangles in the following figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_3fc4f3a6ef7302a792fcg-26.jpg?height=447&width=516&top_left_y=543&top_left_x=646)

In total, there are 27 triangles, each with a perimeter of $3 \mathrm{~cm}$, totaling $27 \cdot 3=81 \mathrm{~cm}$.