To find the height of the rectangular prism, we can use the formula for the volume of a rectangular prism:
\[ V = \text{Base Area} \times \text{Height} \]
Given that the base area of the prism is \(15 , \text{cm}^2\) and we need to find the height, we will use this formula.
We can also note that the volume of the pyramid with a congruent base is given as \(20 , \text{cm}^3\). The formula for the volume of a pyramid is:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Let’s set \(h_p\) as the height of the pyramid. Therefore, we can write:
\[ 20 = \frac{1}{3} \times 15 \times h_p \]
Solving for \(h_p\):
\[ 20 = 5 \times h_p \implies h_p = \frac{20}{5} = 4 , \text{cm} \]
Since the pyramid and prism have congruent bases and a congruent height, the height of the rectangular prism will also be:
\[ h_{\text{prism}} = h_p = 4 , \text{cm} \]
Thus, the height of the rectangular prism is:
\[ \boxed{4 , \text{cm}} \]