In this problem, we are given a right rectangular prism with dimensions $3 \times 4 \times 5$ and asked to find the volume of the octahedron formed by joining the centers of the faces of the prism.

Steps to solve:

1. Understanding the Geometry of the Octahedron: The centers of the faces of the rectangular prism are connected to form an octahedron. This octahedron can be viewed as two identical pyramids joined at their bases, with the base being a rectangle formed by the centers of opposite faces.

2. Finding the Diagonal Lengths: The centers of the faces divide the edges of the prism into half, so the diagonal lengths between opposite centers will be determined by the dimensions of the prism:

   • The distance between the centers of two opposite faces parallel to the $3$-unit side is the diagonal of the $4 \times 5$ rectangle: $\text{Diagonal} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}$
   • The same can be applied for the other faces to find the diagonal distances between the centers of faces parallel to the $4$-unit and $5$-unit sides.

3. Volume of the Octahedron: The volume $V$ of an octahedron can be derived from the formula for a bipyramid, which is given by: $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$ where the base area is the area of the central rectangle formed by the centers, and the height is the distance from the center of one pyramid to its apex (which will be half of the total height of the prism).

Since this geometric transformation of the octahedron from the prism is more complex, we use the fact that the volume of the octahedron is exactly half the volume of the prism when formed in this way.

4. Volume Calculation: The volume of the rectangular prism is: $V_{\text{prism}} = 3 \times 4 \times 5 = 60$ Thus, the volume of the octahedron is: $V_{\text{octahedron}} = \frac{1}{3} \times V_{\text{prism}} = \frac{1}{3} \times 60 = 20$

Final Answer:

The volume of the octahedron is 20 cubic units.

Would you like more details on any step, or have additional questions?

Here are five related questions:

1. How can we generalize the volume of an octahedron formed from any rectangular prism?
2. How are the centers of faces connected to form an octahedron in other polyhedra?
3. What other methods can be used to find the volume of polyhedra from prisms?
4. How is the volume of an octahedron compared to that of other shapes like tetrahedrons?
5. Can you compute the surface area of the octahedron in this example?

Tip: When dealing with complex geometric shapes, symmetry often helps simplify volume and surface area calculations.