3D Mohr's circle conceptual question

[musicmar]

This is more of a general conceptual question than a specific homework problem. I know how to do these problems, but I'm not understanding part of them.
So, with a given stress element, I first look at one specific face, and plot a two-dimensional Mohr's circle. Then, I find the center, radius, and principle stresses. I could also find θ_p, but I'm not sure I need to in this case. If I do, that's what I'm not understanding. I know rotating the circle by 2θ_p and thus the element by θ_p yields a no shear state in the first orientation considered.
My question:
To plot the other two circles, I have drawn the stresses on the remaining two planes of the element. In order for the circles to work, x' and y' (each from one of the planes) must equal the principle stresses. I am confused as to why they don't equal the stresses in their planes. I know it has something to do with the angle of rotation, but I'm not exactly sure what.

An explanation would be greatly appreciated.
Thank you!

 

[Valkarie]

The reason x' and y' don't equal the stresses in their planes is because the Mohr's circle is a representation of the element at two different orientations. When you rotate the element by θp, it changes the orientation of the stress element as well as the orientation of the coordinate system. This means that the same stresses will appear differently in the two orientations. The x' and y' coordinates represent the stresses in the new orientation, which is why they are different from the stresses in the original orientation.