homework and exercises - Why aren't calculation results in error propagation at the center of the range?

We have two copper rods, with $L_1$ and $L_2$ as their lengths respectively, and we want to glue the two bars together, with glue that's infinitesimally thin. $$\begin{align} L_1 &= 20 ± 0.2\ \mathrm{cm} \\ L_2 &= 30 ± 0.5\ \mathrm{cm} \end{align}$$ To calculate the length of the composite bar, $L$, as well as its uncertainty, we can do the following (which I admit is a rather crude method, but is done for completeness): $$\begin{align} L_\text{MAX} &= 20.2 + 30.5 = 50.7\ \mathrm{cm} \\ L_\text{MIN} &= 19.8 + 29.5 = 49.3\ \mathrm{cm} \end{align}$$

Therefore, $L = 50 ± 0.7\ \mathrm{cm}$.

This, although is a long method, is a correct as the length $L$ is just a sum of the values of $L_1$ and $L_2$ with an uncertainty of the range of values possible divided by two.

Now, if we want to calculate the area with the following length and width, as well as the uncertainty, we could use a method similar to the one described above: $$\begin{align} W &= 20 ± 0.2\ \mathrm{cm} \\ L &= 10 ± 0.2\ \mathrm{cm} \\ A_\text{MAX} &= (20.2\ \mathrm{cm})(10.2\ \mathrm{cm}) = 206.04\ \mathrm{cm}^2 \\ A_\text{MIN} &= (19.8\ \mathrm{cm})(9.8\ \mathrm{cm}) = 194.04\ \mathrm{cm}^2 \end{align}$$ In this case, the answer without the uncertainty, $10\ \mathrm{cm} \times 20\ \mathrm{cm} = 200\ \mathrm{cm}^2$, is not the center of our range of values. Although 200 isn't the smack center, there does exist a center, which in this case is 200.04.

The actual uncertainty of the area is 6, which is in fact half of the range of the maximum and minimum, giving us a final answer of $200 ± 6\ \mathrm{cm}$.

The way we have defined the propagation of uncertainties in physics is such that the answer is not necessarily the smack center of the minimum and maximum value range, but is instead the product of the two measurements, the two lengths in this case. This approach to the first problem made a lot of intuitive sense, however, I cannot understand why the final answer is 200 (which is not the center of the range) ± 6, and why this answer gives a different range of values than the range calculated using the long, crude method.

I am a high school student who has not covered calculus yet, which is what prevented me from understanding the proof of adding fractional or percentage uncertainties when we multiply or divide quantities. Any help will be greatly appreciated, thanks in advance.