The area \( A \) of a triangle can be expressed using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the area varies jointly with the height and the base, we can write:
\[ A = k \times \text{base} \times \text{height} \]
where \( k \) is a constant of proportionality.
We know that for one triangle:
- Area \( A = 270 \) square centimeters
- Height \( h = 30 \) centimeters
- Base length \( b = 18 \) centimeters
Substituting these values into the equation, we get:
\[ 270 = k \times 18 \times 30 \]
Calculating \( 18 \times 30 \):
\[ 18 \times 30 = 540 \]
Now we substitute back into the area equation:
\[ 270 = k \times 540 \]
To find \( k \), divide both sides by 540:
\[ k = \frac{270}{540} = \frac{1}{2} \]
Now, we need to find the area of a triangle with:
- Height \( h = 25 \) centimeters
- Base length \( b = 16 \) centimeters
Using the formula for the area with the determined constant \( k \):
\[ A = k \times \text{base} \times \text{height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 16 \times 25 \]
Calculating \( 16 \times 25 \):
\[ 16 \times 25 = 400 \]
Now substituting into the area equation:
\[ A = \frac{1}{2} \times 400 = 200 \]
Thus, the area of the triangle with a height of 25 centimeters and a base length of 16 centimeters is:
\[ \boxed{200} \text{ square centimeters} \]