To find the volume of the pyramid, we use the formula for the volume of a pyramid:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given:
- Volume of the rectangular prism \(V_{\text{prism}} = 420 , \text{in}^3\)
- The base area and height of both the prism and pyramid are the same.
The volume of the rectangular prism is given by:
\[ V_{\text{prism}} = \text{Base Area} \times \text{Height} \]
Since the volume of the prism is 3 times the volume of the pyramid (because the pyramid's volume is \( \frac{1}{3} \) of the prism's volume), we can set up the relationship:
\[ V_{\text{pyramid}} = \frac{1}{3} \times V_{\text{prism}} \]
Now plug in the volume of the prism:
\[ V_{\text{pyramid}} = \frac{1}{3} \times 420 , \text{in}^3 = 140 , \text{in}^3 \]
So, the volume of the pyramid is:
\[ \boxed{140 , \text{in}^3} \]