How are inverse square laws verified? I'm curious about how equations like $F_{g}=G\frac{m_{1}m_{2}}{r^{2}}$ and $F_{e}=k\frac{q_{1}q_{2}}{r^{2}}$ were derived. Turns out that they're actually inverse square laws. But it's still unclear to me, how they are verified.
The equation for moment of inertia is $I=m\sum_{n=1}^{n} r_{n}^{2}$. Does $r^2$ here indicate that rotational inertia is proportional to the area of a... circle or sphere? (Is it a, um... square law?) Which leads me to the last question—
Why are laws like these inversely proportional to the area of a sphere? Couldn't they be inversely proportional to the volume of a sphere? 
Is it even sphere in all cases? 
Why prefer area over volume? 
Most importantly, how are they verified?
I think the answer to this is interrelated to the previous questions. So I asked all of them at once, instead of making seperate posts. Should I make individual posts instead?
PS: Feel free to share other articles, videos, or stackexchange posts related to this topic. 
 A: Since you are asking how an inverse-square law can be verified
(not how it can be derived), the answer is:
You need to do experimental measurements.
For example: Coulomb's law $F_e=k\frac{q_1q_2}{r^2}$
about the force between electrical charges can be verified
as described in Coulomb's law - Simple experiment to verify Coulomb's law.
Basically you take 2 electrical charges, and measure the force
$F_e$ for various distances $r$. Then you calculate
$k = \frac{F_e r^2}{q_1q_2}$
and check that you get the same numerical value $k$ for all
your measurements.

image from Coulomb's law - Simple experiment to verify Coulomb's law 

Deriving an inverse-square law is a completely different story.
It means that you prove that this law follows from another law
which is considered to be more fundamental.
In the example from above, you would start with Gauss' law
for the electric field and the definition of electric field strength
$\mathbf{E}$ (which is $\mathbf{F} = q\mathbf{E}$), do some
clever mathematical conclusions, and finally arrive at Coulomb's law.
A: Short of rewriting a derivation of an inverse square law, I'll just touch on why we'd care about areas rather than volumes. Generally, inverse square laws come from considering the flux of a vector field (for example, an electric field), through a surface. The surface area is involved in these sort of flux integrals, and surface areas go with the square of some length. For example, the surface area of a sphere of radius $r$ is $4 \pi r^2$, and of a cube of side length $a$ is $6 a^2$.
As far as verifying an inverse square law goes, in physics teaching labs, I've verified inverse square laws through linearising experimental data.
Suppose we're trying to measure that the intensity of light measured from a bulb obeys an inverse square law with distance from the bulb. In other words, if $I$ is the intensity and $d$ the distance, that $I = k/d^2$ for some constant k.
Let's take logarithms of whatever base of both sides of this equation, and rearrange to get: $\log(I) = \log(k) - 2 \log(d)$. So, if we plot $\log(I)$ against $\log(d)$, we should get a straight line with $y$-intercept $\log(k)$ (which we don't really care about), and gradient $-2$.
We can verify that this inverse square law by gathering our own data, and comparing it to this prediction. If the gradient of the line in our analysis is $-2$ within error bounds, then we will have verified it.
A: Regarding how they were derived, this might help
http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/isq.html
No, it is not sphere in all cases. It applies to any other non uniform shape.
When you take a certain object with the source of a force in it, for example a positive charge, the force being a vector has a certain direction along the radial line, the radial vector (why? Because it's more general and taking only one of the axes restricts you being only on one of the axes). 
This line passes through the surface of the sphere enclosing the charge and in cases of surfaces you have only area and not volume. 
Suppose you are situated on a point $P$ of this surface, then the force you experience (in our case, a force outwards), will be found on this surface and not the volume. Now suppose you move a bit sideways, then your radial vector would change, and your direction of force will not be same as before. If you were to have a law which is inversely proportional to volume, it would be like saying that force does not depend on direction, which is not true since force is a vector quantity.
[Also mathematically, in case of integration, you write the term 
$da=dxdyk(cap)$
Which denotes the direction in the positive z axis. Volume is given by,
$dxdydz$ 
Which doesn't really have a direction.]
The inverse square law for light intensity can be easily verified by measuring intensities of light at different distances.
