A 4" × 6" × 8" rectangular solid is cut by slicing through the midpoints of three adjacent sides as shown. Find the volume of the outlined solid.
We can use 3d vectors. This pyramid is spanned by the vectors (2,0,0), (0,3,0), and (0,0,4). Taking the determinant gives us a volume of 8.
+32 The area of the triangular pyramid is $\frac{1}{3} bh. $ We can make the base the top face, and the base has an area of $\frac{3 \cdot 4}{2} = 6.$ The height that is given is $\frac{1}{2} \cdot 4 = 2.$ Now, we can plug these values in, and we get $$V = \frac{1}{3} bh. $$ $$V = \frac{1}{3} (6)(2)$$ $$\boxed{V = 4 \text{ in}^3}.$$
