Question
) If a prism and a pyramid have congruent triangular bases and the same heights, then
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GPT-4o mini
If a prism and a pyramid have congruent triangular bases and the same heights, we can compare their volumes.
The volume \( V \) of a prism is given by the formula:
\[
V_{\text{prism}} = B \cdot h
\]
where \( B \) is the area of the base and \( h \) is the height of the prism.
The volume \( V \) of a pyramid is given by the formula:
\[
V_{\text{pyramid}} = \frac{1}{3} B \cdot h
\]
where \( B \) is the area of the base and \( h \) is the height of the pyramid.
Since both the prism and the pyramid have congruent triangular bases and the same height, the area of the base \( B \) is the same for both shapes. Therefore, we can express their volumes in terms of the base area and height.
### Conclusion:
The volume of the prism is three times the volume of the pyramid:
\[
V_{\text{prism}} = 3 \cdot V_{\text{pyramid}}
\]
In summary, if a prism and a pyramid have congruent triangular bases and the same heights, the prism will have a volume that is three times greater than that of the pyramid.
The volume \( V \) of a prism is given by the formula:
\[
V_{\text{prism}} = B \cdot h
\]
where \( B \) is the area of the base and \( h \) is the height of the prism.
The volume \( V \) of a pyramid is given by the formula:
\[
V_{\text{pyramid}} = \frac{1}{3} B \cdot h
\]
where \( B \) is the area of the base and \( h \) is the height of the pyramid.
Since both the prism and the pyramid have congruent triangular bases and the same height, the area of the base \( B \) is the same for both shapes. Therefore, we can express their volumes in terms of the base area and height.
### Conclusion:
The volume of the prism is three times the volume of the pyramid:
\[
V_{\text{prism}} = 3 \cdot V_{\text{pyramid}}
\]
In summary, if a prism and a pyramid have congruent triangular bases and the same heights, the prism will have a volume that is three times greater than that of the pyramid.
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