To find the height of an equilateral triangle, we can use the formula where the height \( h \) equals \(\frac{\sqrt{3}}{2} \times s\), where \( s \) is the side length.
Given \( s = 8 \) meters and using \(\frac{7}{4}\) for \(\sqrt{3}\), we can substitute the values into the formula:
\[ h = \frac{\sqrt{3}}{2} \times s = \frac{7/4}{2} \times 8 \]
First, simplify \(\frac{7/4}{2}\):
\[ \frac{7/4}{2} = \frac{7/4}{2/1} = \frac{7}{4} \times \frac{1}{2} = \frac{7}{8} \]
Now, multiply by the side length:
\[ h = \frac{7}{8} \times 8 = 7 \text{ meters} \]
Therefore, the approximate height of the equilateral triangle with a side length of 8 meters is \(\boxed{7}\) meters.