If the circle moves in the direction perpendicular to the plane determined by itself, there would be no Lorentz contraction.

So you have to consider only the case when the velocity is in the plane where the circle lies. In this case, Lorentz contraction happens in the direction of the velocity but not in the direction perpendicular to the velocity, so the circle is an oval as viewed by an observer at rest. So the area changes by the same factor for a line.

Let's call it the "simultaneous cross section" that tends to zero area as the hoop is lorentz contracted along one dimension.

The shape is an ellipse with semimajor axis equal to the original radisu, and the semiminor axis equal to the lorentz contracted radius.

Actually, the question that brought this about is, "does the irrational number π ≈ 3.14159 remain a universal constant for such travel?" Based on the answers above, for the observer at rest π ceases to be constant, if calculated using the area of such a circle divided by the square of its radius, which is its actual definition. Could one then conclude that π is not itself universally constant?

Well, Pi is essentially the area of a circle with radius 1.

A circle is defined as being the locus of points in a plane equidistant from a point. A point is generally regarded as "that which has no part" I think it was defined in Euclid's Elements.

However, a point actually does have a part. It's got a velocity.

If that velocity is zero, then the ratio between the area and radius is Pi. If that velocity is a significant portion of the speed of light, it becomes obviously different from Pi.

To understand how an area transforms under Lorentz transformations is well known-it's called the Nambu-Goto action, that generalizes the proper time of a particle. Cf. https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Nambu-Goto_action.html

So a (space-like) circle can be defined by the coordinates: X^0=0, X^1=rcosθ, X^2=rsinθ, X^3=0, with 0≤r≤R and 0≤θ≤2π. r and θ are the coordinates of the surface, bounded by the circle, the X's describe how it's embedded in spacetime. It suffices to do a Lorentz transformation in the X's and recompute the area. Similar reasoning holds for volumes.

Standard exercises in multivariable calculus.

R.O. : the answer to your "actual question" of course depends on what you call the radius of an ellipse. If it's the geometric mean between major and minor axes, your pi is still 3.14....