Volume question HARD

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.

Side lengths are half of each side so 3*2*4=24/2=12

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 $$V = \frac{1}{3} bh.$$ $$V = \frac{1}{3} (6)(2)$$ $$\boxed{V = 4 \text{ in}^3}.$$

Question: If a blue triangle is the base of a pyramid, then what is its height?