For an equilateral triangle, all three sides are of equal length, and the perimeter is the sum of these three sides.

Step 1: Find the length of one side

Given the perimeter is 36 dm, and since all sides are equal, each side is: $\text{Side length} = \frac{36 \, \text{dm}}{3} = 12 \, \text{dm}$

Step 2: Calculate the area of the equilateral triangle

The formula for the area $A$ of an equilateral triangle with side length $s$ is: $A = \frac{\sqrt{3}}{4} \times s^2$

Substitute $s = 12 \, \text{dm}$: $A = \frac{\sqrt{3}}{4} \times (12)^2 = \frac{\sqrt{3}}{4} \times 144$ $A = 36\sqrt{3} \, \text{dm}^2$

To approximate the area numerically, use $\sqrt{3} \approx 1.732$: $A \approx 36 \times 1.732 = 62.352 \, \text{dm}^2$

Final Answer:

The area of the equilateral triangle is approximately $62.35 \, \text{dm}^2$.

Would you like further details or have any questions?

Here are five related questions:

1. What is the height of the equilateral triangle?
2. How does the formula for the area of an equilateral triangle differ from other triangles?
3. Can you calculate the area of an equilateral triangle with a different perimeter?
4. What is the significance of the $\sqrt{3}$ in the area formula for equilateral triangles?
5. How does the perimeter relate to the side lengths in different types of triangles?

Tip: For any regular polygon, the area formula can often be derived from symmetry and trigonometry, especially for triangles and hexagons.