To find the height of the cone, we can use the volume formula for a cone, which is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where:
- \( V \) is the volume of the cone,
- \( r \) is the radius,
- \( h \) is the height.
From your question, we know:
- The volume \( V = 225\pi \)
- The radius \( r = 5 \)
Now we can plug in the values into the volume formula:
\[ 225\pi = \frac{1}{3} \pi (5^2) h \]
First, calculate \( 5^2 \):
\[ 5^2 = 25 \]
Substituting back into the equation gives:
\[ 225\pi = \frac{1}{3} \pi (25) h \]
Next, simplify the right side:
\[ 225\pi = \frac{25}{3} \pi h \]
Now, we can eliminate \( \pi \) from both sides:
\[ 225 = \frac{25}{3} h \]
To solve for \( h \), multiply both sides by 3:
\[ 675 = 25h \]
Now divide both sides by 25:
\[ h = \frac{675}{25} = 27 \]
Thus, the height of the cone is:
\[ \boxed{27} \]