Question
One of Mr. Newman’s architecture students states that prisms and pyramids are different shapes, but the volume of a pyramid is half the volume of a prism with the same dimensions. Another student argues that the volume of a pyramid is one-third the volume of a prism with the same dimensions. Who is correct? Explain.
Answers
GPT-4o mini
The first student is correct. The volume of a pyramid is indeed one-third the volume of a prism (specifically, a rectangular prism) with the same base area and height.
Here’s why:
1. **Volume of a Prism**: The volume \( V \) of a prism is calculated using the formula:
\[
V = \text{Base Area} \times \text{Height}
\]
where the base area is the area of the base shape (e.g., rectangle, triangle) and height is the perpendicular distance between the two bases.
2. **Volume of a Pyramid**: The volume \( V \) of a pyramid is calculated using the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The base area is the same as in the prism, and the height is also the same as the height of the prism.
To compare them:
- For a prism, the volume is \( V_{\text{prism}} = \text{Base Area} \times \text{Height} \).
- For a pyramid, the volume is \( V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).
Therefore, the volumetric relationship shows that the volume of a pyramid is one-third that of a prism with the same base area and height, not half.
In summary, the correct statement is that the volume of a pyramid is one-third the volume of a prism with the same dimensions. Thus, the first student was mistaken in saying it's half; the other student was correct in stating that it is one-third.
Here’s why:
1. **Volume of a Prism**: The volume \( V \) of a prism is calculated using the formula:
\[
V = \text{Base Area} \times \text{Height}
\]
where the base area is the area of the base shape (e.g., rectangle, triangle) and height is the perpendicular distance between the two bases.
2. **Volume of a Pyramid**: The volume \( V \) of a pyramid is calculated using the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The base area is the same as in the prism, and the height is also the same as the height of the prism.
To compare them:
- For a prism, the volume is \( V_{\text{prism}} = \text{Base Area} \times \text{Height} \).
- For a pyramid, the volume is \( V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).
Therefore, the volumetric relationship shows that the volume of a pyramid is one-third that of a prism with the same base area and height, not half.
In summary, the correct statement is that the volume of a pyramid is one-third the volume of a prism with the same dimensions. Thus, the first student was mistaken in saying it's half; the other student was correct in stating that it is one-third.