To find the height of the cone, we can use the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height.
Given that the volume \( V = 225\pi \) cubic meters and the diameter of the cone is 10 meters, we first find the radius:
\[ r = \frac{\text{diameter}}{2} = \frac{10}{2} = 5 \text{ meters} \]
Now we can substitute the values into the volume formula:
\[ 225\pi = \frac{1}{3} \pi (5^2) h \]
Calculating \( 5^2 \):
\[ 5^2 = 25 \]
Substituting back into the volume equation gives:
\[ 225\pi = \frac{1}{3} \pi (25) h \]
We can simplify this by dividing both sides by \( \pi \):
\[ 225 = \frac{1}{3} (25) h \]
Now, multiply both sides by 3 to eliminate the fraction:
\[ 675 = 25h \]
Now, solve for \( h \) by dividing both sides by 25:
\[ h = \frac{675}{25} = 27 \]
Thus, the height of the cone is
\[ \boxed{27} \text{ meters} \]