Why is the universe expanding yet remaining flat? According to The Big Bang Theory everything started from nothing and suddenly started expanding creating everything that is. So why is it only expanding on an X&Y axis? I can only find facts that point towards it expanding in radius, but never thickness. Is it doing this as well, but I have just happened to never find it? Or is there some prevailing theory that has eluded me that explains this?
 A: It seems that you are interpreting the Universe as a plane layer with a given thickness. That is not true, the Universe is isotropic (it looks the same in all directions) and so is its expansion. When cosmologists say the Universe is flat they mean that at a large-scale it has a vanishing curvature. A way to understand this without knowing about differential geometry is to imagine a space where any (large scale) triangle has its internal angle added up to $180^o$.
A: In special relativity, the geometry of spacetime is described by $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=c^2dt^2-dx^2-dy^2-dz^2=c^2dt^2-dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2.$$(Note the space part can be described with Cartesian or polar coordinates.) This works well on small scales, but to understand the Big Bang we have to go beyond this.
Regardless of how if at all the Universe "began", the time-dependence of the universe's size follows from adding to general relativity two assumptions, isotropy (all directions are equivalent rather than there being an "up/down" axis across space) and homogeneity (on a large scale the universe has uniform density). One can then prove $$ds^2=c^2dt^2-a^2\left( t\right)\left(\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\phi^2\right)$$for some constant $k$ and some function $a$ called the scale factor. It is $a$ that determines the size of the universe, and it obeys two differential equations that govern its change over time. A "flat" universe is just one where $k=0$, or at least where $k$ is so small we can't measure how it differs from zero.
A: Think of it as a plane sheet, and expansion is simply the sheet growing in size (or you pulling the opposite edges apart, and let's assume the sheet does NOT break, it just expands) it stays a sheet, i.e. flat. Two point drawn on the sheet will get further apart.
If it was a positive curvature it'd like a balloon's surface, and you blow on it air and it expands: two points drawn on the balloon surface will get further apart. The same is true for a negative curvature saddle type surface expanding
But we do know that the universe, on a large scale (sizes of clusters of galaxies, tens to hundred light years and more), is flat (and homogeneous), or very close (uncertainty of .4%). So it is that flat sheet.
Still, remember we are talking about the spatial parts of the universe (at each time, whatever the space looks like), which are 3D (3 dimensional). So think of the sheet as a 3D sheet. That's the way to visualize it. The math is the answer you already accepted which shows the metric, and flat means k=o, so the spatial parts (called spatial hypersurfaces, but it's just the space we look at) are flat. We've measured k to be 0 within a .4% margin. K =1 is positive curvature, K=-1 is negative, and any other k can be mathematically reduced to one of those, just whether zero, positive or negative. a(t) in that same equation is simply the scale of the size of the sheet which is expanding uniformly in all dimensions. 
So it's well defined, well understood, well visualized, and well measured. 
A: It is expaning in x,y and z directions. The radius of a sphere is $r=\sqrt{x^2+y^2+z^2}$. I'm guessing maybe you confused this with the formula for radius in 2D $r=\sqrt{x^2+y^2}$.
A: To add to some other answers, it is possible for a surface to be intrinsically flat but also to be closed. A ball has positive curvature, but a ring like a donut or a life preserver has zero overall curvature, with the locally negative bits exactly cancelling the locally positive ones; you can even divide it up into a neat checkerboard. If you imagine a small rubber life preserver being inflated, it will steadily expand while its intrinsic metric remains flat. The universe is a 3D version of that.
For a pretty much math-free account of the geometric ideas involved, see:


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*Jeffrey R. Weeks; The Shape of Space, CRC Press, 2002.

