Why are measurements standardized the way they are?

Using meters as a base length, squaring or cubing lengths smaller than 0.67m makes the square term larger than the cubed term. This fact causes certain properties of physics (how rain needs to form?) to take place, right?

So what if the standard length was not meters, but some smaller unit. Squaring 0.5m gives a larger result than cubing, but with a smaller base unit it would be smaller.

I don't know if I have explained this properly, but I feel like I'm missing something about how units of measurement are used.

• Sorry, but I just can't get your point. As long you state, say squared or cubed metres after the figure, I don't see what the problem is. – user81619 Nov 20 '15 at 1:21
• You cannot say that $(0.67\ \mathrm{m})^2$ is smaller than $0.67\ \mathrm{m}$, because they have different units. – Javier Nov 20 '15 at 1:27
• Standardised of measurement, like the meter or the kilogram, are essentially conventionally agreed on units against which everything else can be measured. That choice does not affect physics itself though. Large parts of the Anglo-Saxon engineering literature still uses foot and pound: the science described therein is no different from that described using S.I. units. – Gert Nov 20 '15 at 1:27
• Hello, and welcome to Stack Exchange. No matter what length you choose as "one unit", smaller distances squared or cubed will result in even smaller numbers. Unless you chose the Planck length, in which case you'd have extremely unwieldy numbers. – Daniel Griscom Nov 20 '15 at 2:06
• @DanielGriscom How would using the Planck length change this? I can work in units of 0.1 Planck length if I want and it's still true that the number I get when I cube lengths smaller than one of those units is smaller than the number I cubed. – DanielSank Nov 20 '15 at 2:13

Physics processes are independent on the measurement units that you use. Your question is not about physics but about math. If you square a number less than one it will result in a smaller number, if you square a number larger than one you will get a larger number. But changing units will not change anything. For instance, suppose you have a square that is 0.5 meter on a side, then the area will be $0.25 m^2$. If you measure the same square in centimeters, the side will be 50 cm long, and the area will be $2500 cm^2$. But both the lengths and areas are still the same: $50cm=0.5m$ and $0.25 m^2=2500 cm^2$

I'm not sure about your rain comment and how you were taught rain formed, but I can give you a bit of info on units.

It doesn't matter what length you choose to be the base unit, as long as you are consistent. Right now our definition of the meter is the length light travels in $1/c$ seconds, where $c$ is the defined speed of light.

You have to remember that physical constants will scale when you change units, and physics equations are usually unit independent.

Why are measurements standardized the way they are?

This is a really smart question.

France used many units of measurement before 1789. The new government that made a lot of reforms decided also to impose meter and kg on the entire territory of France. Simply, the weight of a cube of water (a substance which is easily available) was considered to be 1 kgf and each of the cube sides 0.1 m.