2015 AMC 10B Problem 17

The octahedron can be viewed as two congruent pyramids whose shared base is the rhombus through the centers of the four side faces. This rhombus has diagonals $4$ and $5$, so its area is $\frac12\cdot4\cdot5=10$. Each pyramid has height $\frac32$, half the prism’s height. Thus the total volume is $$2\cdot\frac13\cdot10\cdot\frac32=10.$$ Thus, the correct answer is B.