The intuition behind the parametrization of a triangular prism

Question

I'm trying to understand how the shape of a prism is defined. Imagine a triangular prism, with ends given by the planes $y = 0$ and $y = 2$ and remaining faces given by the planes $x = 0$, $z = 0$ and $x + 4z = 4$

The end goal of the problem to find the volume $V$, computationally that's not a problem for me. However, the define geometry is.

Why is the second point of $z$ parameterized as $(x + 4z = 4)$ or $ (z = \frac{4-x}{4})$

Also, I have an issue with imagining the defined points, the single varable points(such as $y_i=0$, and $y_f=2$) can be dots joined by a boundary line, yet combined with the other points(such as $z$ & $x$)they would define a surface.

The difference between a point, boundary line, surface is a bit confusing.

The geometric intution for me is difficult in comparison to the direct computation of $V=\int\int\int_v dV$

Answer 1

You keep referring to "points" when the objects under discussion are not points, but planes:

1. "$y=0$" is the equation of the plane with points $\{(x,0,z) \mid x, z \in \Bbb R\}$. This is the plane containing the $x$ and $z$ axes.
2. "$y=2$ is the equation of the plane with points $\{(x,2,z) \mid x, z \in \Bbb R\}$, which is parallel to the previous plane and $2$ units away from it in the positive $y$ directions.
3. "$x=0$" is the equation of the plane with points $\{(0,y,z) \mid y, z \in \Bbb R\}$. This is the plane containing the $y$ and $z$ axes.
4. "$z=0$" is the equation of the plane with points $\{(x,y,0) \mid x, y \in \Bbb R\}$. This is the plane containing the $x$ and $y$ axes.
5. "$x+4z = 4$" is the equation of the plane with points $\{(4 - 4z,y,z) \mid y, z \in \Bbb R\}$. This plane is parallel to the $y$-axis, but is slanted in the $x$ and $z$ directions.

The prism is the region that lies inside all of these planes. The region that is bounded by the planes on all sides.