Transformers have a number of technically relevant properties that make each transformer design fairly unique. Often these parts have to be engineered for a particular application. Let's discuss a few of the most important aspects of transformer design.
From an engineering perspective a transformer has to work at a certain frequency (or a range of frequencies), it has to be able to transfer a certain amount of power and it has to have a desired voltage ratio between any two pairs of its winding.
The amount of power a transformer can transfer depends primarily on the magnetic properties and the volume of its core. This power is frequency dependent and it scales essentially with $P\propto V\times f$. Transformer design therefor has to start with the choice of a core of sufficient size that can transfer the required amount of power at the design frequency.
For a given core size and core material (which also depends on the application) the number of turns then determines the inductance and the capacitance of the primary and secondary coils. This, in turn, determines the useful impedance and frequency range of the transformer.
To make life easy for the designer, the first important figure of merit of a transformer core is called the $A_L$ value, which is usually specified with units of $\mathrm{nH}/n^2$. The inductance of a winding with $n$ turns on a core with $A_L$ is given by the formula $L=n^2A_L$. $A_L$ basically assumes that the magnetization of the core material is linear and it depends on the size and shape of the core, already, so that we don't have to calculate the effective magnetic flux area and effective magnetic path length for a given core size. As a rule of thumb, larger cores (of the same shape) made of the same material have larger $A_L$ values (because the area scales quadratically with size while the magnetic flux length only increases linearly).
Here are three examples of how this works out in practice:
Example 1: RF transformer
A primary winding on a core with $A_L=100\:\mathrm{nH}/n^2$ and $1$ turn will have an inductance of $100\:\mathrm{nH}$. This single turn transformer will only be useful for high frequencies in the $>100\:\mathrm{MHz}$ range. At the lower end of its useful design frequency (for use in typical $50\Omega$ systems) the effective impedance of the primary winding will be
$$Z=2\pi\times f\times L = 2\pi \times 100\:\mathrm{MHz}\times 100\:\mathrm{nH}=62.8\:\mathrm \Omega.$$
This is suitable as an RF transformer in systems with $50\:\mathrm\Omega$ impedance but can't be used at much lower frequencies.
Windings have both an inductance and a self-capacitance. In case of this single turn RF transformer, we may have a typical winding capacitance of approx. $1\:\mathrm{pF}$. If we insert this into the formula for an LC resonance circuit, we find a self-resonance frequency of
$f_{Self-Resonance}={1\over{2\pi\sqrt{LC}}} = {1\over{2\pi\sqrt{10^{-12}\mathrm F\times 10^{-7}\mathrm H}}}\approx 500\:\mathrm{MHz}$.
Our RF transformer is therefore limited to a useful operating range of $100\:\mathrm{MHz}$ at the low end because of its inductance and $500\:\mathrm{MHz}$ at the high end because of its self-resonance frequency. With careful design techniques this range can be improved quite a bit, but RF transformers rarely have wider bandwidth ranges than $1:100$ and many work best over no more than a couple of octaves.
Example 2: Switching power supply transformer
The same core wound with $50$ turns will have an inductance of $$L=50^2\times100\:\mathrm{nH}/n^2=2500\times100\:\mathrm{nH}=250000\:\mathrm{nH}=250\mathrm{\mu H}.$$
Because the design with a $50$-turn primary winding has an inductance that is 2500 times higher, it will perform well in applications that are running at frequencies 1000 times lower than our first example and is therefor useful at frequencies of around $100\:\mathrm {kHz}$. Such a transformer will, for instance, be found in switching voltage converters, which are typically operating in the $50kHz-4MHz$ range.
Because a larger number of turns means that we have to use thinner wires with thinner insulation, the medium frequency transformer with its $50$ turns on the primary has a much higher winding capacitance (in the range of $10\:\mathrm{pF}$ to hundreds of $\mathrm{pF}$, depending on how much care is put into the winding scheme), which means it has a much lower self-resonance frequency. Technically useful designs will have self-resonance frequencies around the $10\:\mathrm{MHz}$ range.
Example 3: Audio transformer
If we want to build transformers for much lower frequencies, then we need cores with much higher $A_L$ values (e.g. $5\:\mathrm{\mu H}/n^2$) and we will need to add hundreds or thousands of turns. A 1000-turn transformer on a $A_L=5\:\mathrm{\mu H}/n^2$ core will have an inductance of $L=10^6\times 5\:\mathrm{\mu H}=5\:\mathrm{H}$. This transformer will have an impedance of $Z\approx 600\:\mathrm \Omega$ at $20\:\mathrm{Hz}$ and will typically be used in audio amplifiers.
The $1000$ turn audio transformer, on the other hand, will perform over a range of approx. $15\:\mathrm{Hz}$-$25\:\mathrm{kHz}$ if wound really well, but there is some art and manufacturing know-how to making these wide-band transformers. A poorly calculated, poorly wound device will not work well at all in audio applications.
There are additional considerations that limit the performance of a transformer. For applications which transfer significant amounts of power one also has to take the resistance of the winding and the skin effect into account, both of which lead to $I^2R$ losses and heating. Additional losses are caused by the hysteresis of the core material's magnetization curve and the eddy currents that can be induced in electrically conductive core materials like transformer steel and some ferrites. These core losses have to be carefully considered during material selection and they are also core shape dependent, which leads to a great number of core geometries for different applications.
Designing a high quality transformer requires that the design engineer picks the right core (shape, material and size) and that the correct wire diameter and number of turns for each winding are used. These devices also have to be tested for their performance before being used in a circuit. For some transformers (like $50/60\:\mathrm{Hz}$ power transformers) this is comparatively easy, but for signal and especially wide-band RF transformers often a significant amount of iterative optimization is needed.
Best Answer
The original engineer may have been assuming a square crossection. Yes, for DC anyway, the resistance of a wire is inversely proportional to the area of its corssection.
By the way, your resistivity of 2.82 x 10-8 can't possibly be right, as should be obvious from a dimensional analisys alone.