Question
I know that rectangular prisms and pyramids have congruent bases and heights. If the volume of a prism is 420 in3, what is the volume of the pyramid?
Answers
GPT-4o mini
To find the volume of a pyramid that has the same base area and height as a rectangular prism, you can use the relationship between the volumes of the two shapes. The volume \( V \) of a rectangular prism is given by the formula:
\[
V = \text{Base Area} \times \text{Height}
\]
The volume \( V \) of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Given that the volume of the prism is 420 in³, we can express the volume of the pyramid as:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} V_{\text{prism}}
\]
Substituting the volume of the prism:
\[
V_{\text{pyramid}} = \frac{1}{3} \times 420 \text{ in}^3
\]
Calculating this gives:
\[
V_{\text{pyramid}} = \frac{420}{3} = 140 \text{ in}^3
\]
Therefore, the volume of the pyramid is \( \boxed{140} \) in³.
\[
V = \text{Base Area} \times \text{Height}
\]
The volume \( V \) of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Given that the volume of the prism is 420 in³, we can express the volume of the pyramid as:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} V_{\text{prism}}
\]
Substituting the volume of the prism:
\[
V_{\text{pyramid}} = \frac{1}{3} \times 420 \text{ in}^3
\]
Calculating this gives:
\[
V_{\text{pyramid}} = \frac{420}{3} = 140 \text{ in}^3
\]
Therefore, the volume of the pyramid is \( \boxed{140} \) in³.