We learn in linear algebra, or even in analytical geometry that given two vectors $\vec{u}$ and $\vec{v}$ the area of the parallelogram formed by them is given by $A=||\vec{u}\times\vec{v}||$, and even more, given a third vetor $\vec{w}$ the volume of the paralelepipedal is given by $V=\langle\vec{w},\vec{u}\times\vec{v}\rangle$, we can scale the area to that of a triangle using a factor of $\frac{1}{2}$ and the volume follows similar logic.
My question is, how do I find this scaling constant to find the areas and volumes of others politopes? Like a pentagon, hexagon, ...