To calculate the height of an equilateral triangle with a perimeter of 54 cm, we can follow these steps:

Step 1: Find the side length of the equilateral triangle.

Since the perimeter of an equilateral triangle is the sum of all its sides, and all sides are equal, the side length $s$ is:

$\text{Side length} = \frac{\text{Perimeter}}{3} = \frac{54 \, \text{cm}}{3} = 18 \, \text{cm}$

Step 2: Use the formula for the height of an equilateral triangle.

The height $h$ of an equilateral triangle can be calculated using the formula:

$h = \frac{\sqrt{3}}{2} \times s$

Substituting $s = 18 \, \text{cm}$:

$h = \frac{\sqrt{3}}{2} \times 18 = 9\sqrt{3} \, \text{cm}$

Approximating $\sqrt{3} \approx 1.732$:

$h \approx 9 \times 1.732 = 15.588 \, \text{cm}$

Final Answer:

The height of the equilateral triangle is approximately $15.59 \, \text{cm}$.

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Would you like more details on how this formula works, or do you have any other questions? Here are some related questions:

1. How is the area of an equilateral triangle calculated?
2. What is the relationship between the height and side length of a triangle?
3. Can the height formula for an equilateral triangle be used for isosceles triangles?
4. How do you calculate the perimeter of an equilateral triangle if you know the height?
5. What happens to the height if the perimeter of the equilateral triangle changes?

Tip: Remember that in an equilateral triangle, the height divides the triangle into two 30-60-90 right triangles, which makes using the formula for the height easier.