Question

Given a triangular pyramid and a triangular prism with congruent bases and the same heights, what do you know about the relationship between the volume of each? Write your answer in fraction form.

Answer 1

To compare the volumes of a triangular pyramid and a triangular prism that have congruent bases and the same heights, we can use the formulas for the volumes of each solid.

1. **Volume of the Triangular Prism**:
The volume \( V_p \) of a triangular prism is given by the formula:
\[
V_p = B \cdot h
\]
where \( B \) is the area of the triangular base and \( h \) is the height of the prism.

2. **Volume of the Triangular Pyramid**:
The volume \( V_t \) of a triangular pyramid (tetrahedron) is given by the formula:
\[
V_t = \frac{1}{3} B \cdot h
\]
where \( B \) is the area of the triangular base and \( h \) is the height of the pyramid.

From these formulas, we can see the relationship between the volumes:

The volume of the triangular prism is:
\[
V_p = B \cdot h
\]

The volume of the triangular pyramid is:
\[
V_t = \frac{1}{3} B \cdot h
\]

Now, we can relate the two volumes:
\[
\frac{V_t}{V_p} = \frac{\frac{1}{3} B \cdot h}{B \cdot h} = \frac{1}{3}
\]

Thus, the volume of the triangular pyramid is \(\frac{1}{3}\) the volume of the triangular prism.

In fraction form, the relationship is:
\[
\frac{V_t}{V_p} = \frac{1}{3}
\]