To find the height of an equilateral triangle given the side length, we can use the formula:
\[ \text{Height} = \frac{\sqrt{3}}{2} \times \text{side length} \]
In this case, we are given the side length of the triangle as 12 meters, and we will use \( \frac{7}{4} \) as an approximation for \( \sqrt{3} \).
First, we substitute the values into the formula:
\[ \text{Height} = \frac{7}{4} \times \frac{1}{2} \times 12 \]
Calculating \( \frac{1}{2} \times 12 \):
\[ \frac{1}{2} \times 12 = 6 \]
Now substituting back into the height formula:
\[ \text{Height} = \frac{7}{4} \times 6 \]
To compute \( \frac{7}{4} \times 6 \), we can rewrite \( 6 \) as \( \frac{6}{1} \):
\[ \text{Height} = \frac{7 \times 6}{4 \times 1} = \frac{42}{4} \]
Now simplifying \( \frac{42}{4} \):
\[ \frac{42}{4} = 10.5 \]
Therefore, the approximate height of the equilateral triangle with a side length of 12 meters is
\[ \boxed{10.5} \text{ meters}. \]