I’ve been trying to solve a problem based on the shadow area of a cube and I want to find the relationship between angles between the square faces of a cube.
In this case, the angle $\theta$ is between the normal to a face of the cube and the plane.
I know the angle between two faces -f1 and f2- is $\pi/2$ So if one shadow is defined by $ s^2\cos \theta $ then the other will be $s^2\sin \theta$
Using this same logic the relationship between f1 and f3 will be the same, but that leads to the shadows of f2 and f3 both being $s^2 \sin\theta$ Which doesn’t make sense since they are also right angles to each other
I want to model the relationship between these angles and make a combined equation adding them up. My equation for the shadow area of a square is $$ s^2 \sin\theta\cos\theta $$ where $s$ is the side length
If I add up the areas of three faces joined by a single vertex I’ll get the shadow area of the cube but for that, I need to find the relationship between the angles.
The equation I found is also shown in this post based on the same question but slightly different.
Shadow area of cube assuming parallel light rays
The reason you cannot reconcile the shadow areas of the other two faces is that the shadow of one face does not determine the shadow of the adjacent faces. This is not a problem if all you want is an average area, but for your purpose the angle $\theta$ is not enough information.
If you use two angles as originally proposed in The average area of the shadow of a square, you can describe the orientation of the cube sufficiently to get areas of three adjacent sides.
You can start with a cube centered at the origin of a three-dimensional Cartesian coordinate system, with the edges of the cube parallel to the coordinate axes. Rotate the cube around the $x$ coordinate axis by $\theta_x$ first, then rotate around the $y$ coordinate axis (not an axis of the rotated cube) by $\theta_y$. Then project the shadow of the cube orthogonally onto some plane parallel to the $x,y$ plane.
After the first rotation, the face originally on the bottom (face $f_1$) will have a shadow of area $s^2 \cos\theta_x$. One of the adjacent faces (face $f_2$) now has a shadow of area $s^2 \sin\theta_x$. Another face (face $f_3$) adjacent to both the faces $f_1$ and $f_2$ has a shadow of area zero, because its plane is still perpendicular to the $x,y$ plane.
The second rotation reduces the areas of the shadows of faces $f_1$ and $f_2$: the shadow of face $f_1$ now has area $s^2 \cos\theta_x\cos\theta_y$ and the shadow of face $f_2$ now has area $s^2 \sin\theta_x\cos\theta_y$. The shadow of face $f_3$ is increased to $s^2 \sin\theta_y$.
An alternative model would be to rotate first by angle $\theta_z$ around the $z$ axis, then $\theta_x$ around the $x$ axis. Then face $f_1$ has area $s^2 \cos\theta_x$, so $\theta_x$ is like your angle $\theta$. The angle $\theta_z$ determines the areas of faces $f_2$ and $f_3$, which vary between $0$ and $s^2 \sin\theta_x$, but in opposite directions (as $\theta_z$ increases, one face's shadow's area increases while the other decreases).