What is the relationship between the volumes of a pyramid and a prism that have the same base area and height?

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The relationship between the volumes of a pyramid and a prism that have the same base area and height can be described using the formulas for their volumes.

1. **Volume of a Prism**: The volume $$ V $$ of a prism is given by the formula:
   $$
   V_{\text{prism}} = \text{Base Area} \times \text{Height}
   $$

2. **Volume of a Pyramid**: The volume $$ V $$ of a pyramid is given by the formula:
   $$
   V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
   $$

If both the pyramid and the prism have the same base area $$ B $$ and the same height $$ h $$, we can express their volumes as follows:

- Volume of the prism:
  $$
  V_{\text{prism}} = B \times h
  $$

- Volume of the pyramid:
  $$
  V_{\text{pyramid}} = \frac{1}{3} \times B \times h
  $$

From these formulas, we can see that the volume of the pyramid is one-third of the volume of the prism. Therefore, the relationship can be summarized as:

$$
V_{\text{pyramid}} = \frac{1}{3} V_{\text{prism}}
$$

In conclusion, for a pyramid and a prism with the same base area and height, the volume of the pyramid is one-third the volume of the prism.
The volume of a pyramid is one-third the volume of a prism with the same base area and height.

What is the relationship between the volumes of a pyramid and a prism that have the same base area and height?

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