Length, Area, Volume - Do We Need Them?

An important tool one can use to understand different concepts is the question, "Do we need them?" Here, by "we", I mean all of humanity as a whole.
That is, can you think of a scenario outside of the examination room where a person would feel the need to invent the concept in question? Of course, if a concept exists in the text books of today, the first person who invented it could not have possibly been motivated by doing good in an exam that tested that concept.
It is often useful to imagine yourself in that person's shoes and think of the reason that led him to the discovery. Let me demonstrate.
Volume.

We all know that volume of a cuboid is length times breadth times height and volume of a sphere is four-thirds of pi times its radius cubed.

But do we really need to define this quantity called volume? Do we really need all these formulas to compute it? Let's put ourselves in the shoes of a possible inventor of the concept. Imagine you live in a world where no one knows how to compute the volume of different objects.

The ruler of your kingdom is a man who takes interest in art and architecture. He visits his friend in Egypt and is fascinated by the pyramids.

On his return, he summons you to the court and orders you to build an exact replica in your own kingdom. He also informs you that his good Egyptian friend has kindly agreed to supply you with one million bricks.

Are one million bricks enough to build an exact replica? It is clear that to answer such a question, you need to calculate the volume of the pyramid, the total volume of all the bricks combined and see which one is bigger.

So, here we go. Here is one situation where you really need a formula for volume.

In general, volume is a quantity that is preserved under deformations of shape, as long as the deformations do not include compression and expansion. For example, if you take a huge object, chip out a small portion of it and reattach it at another location, the total volume of the object will not change.
Thus the question, "Can you deform an object in some way so that you end up with another given object?" can be answered by calculating the volumes of the two objects and comparing them.

So given a large piece of clay, if you are asked to build an exact replica of an object out of it, you should compare the volumes and see if it is possible.
Alright, it's clear now that we do need something, some kind of formula that puts a number on an object so that when we compare the number on one object and the number on another object, we can figure out if one can be deformed without expansion or compression to get the other.

But do we really need the specific formulas we are taught in school? That is, are they really the unique solution to the problem of deciding whether one object can be deformed to get another? Let's think of that formula in an abstract way for a while and let's say the formula is a person and let's call him The Volume God.
Whenever you want to calculate the volume of an object, you show it to The Volume God.
He inspects it for a while and puts a number on it with the Holy Stamp. This number denotes the volume of the object.
Thus, given some clay, and given an object, if you want to build an exact replica of the object, you show the clay and the object to The Volume God, get them both stamped and compare the two numbers. If the number on the clay is bigger, you go ahead and build the object otherwise you complain to your king and ask for more clay.
One day, The Volume God feels slightly mischievous and decides to confuse his people. He decides that instead of putting down the correct volume, next time he is shown an object, he is going to add 10 to the correct volume and stamp it with that number instead.

Thus when shown a cuboid, he started stamping it with 10 + (length x breadth x height) instead of just length x breadth x height. Of course, people had full faith in him and his Holy Stamp and so no one doubted the number.
Surprisingly though, nothing changed.
The Volume God's mischievous plans had absolutely no effect on the people.

The reason is that no one was ever interested in the absolute number that The Volume God stamped on objects. People were only interested incomparisons between two numbers and the comparison never changed.

If a number was less than another, 10 added to the first one was also less than 10 added to the second one. But wait a second. Doesn't this mean that we don't really need the exact formulas that we are taught in school? That if we add some arbitrary number to all the formulas, as long as the number added is consistently the same, we will never know the difference in the real world? Perhaps the volume of a cuboid should have been 10 + (length x breadth x height) then and the volume of a sphere should have been 10 + (4/3 x pi x radius cubed)! All these conclusions are correct if all we ever do with volumes is compare them.
However, we do use volumes for more than just comparisons.
Let's go back to the example of building objects out of clay.

Suppose you are shown an object and a bucket of clay.
Your clay supplier tells you that he can provide you with as many buckets of clay as you want to build an exact replica of the object. How many buckets should you order? You can answer this by getting more demanding towards The Volume God. So you go and pray to him thusly: "It would be great if you could write numbers on these objects such that just by looking at the numbers not only can I figure out if the second object can be deformed into the first, but also, how many of the second objects I would need to build the first object.
Amen." The Volume God agrees and stops adding 10 to the numbers. So now, to decide how many buckets of clay you will need to build an object, you show the object and the bucket to The Volume God and divide the number on the object by the number on the bucket.
That's it. All your problems are solved and the world lives happily ever after.
Wait.
Not ever after.

After only a few days, The Volume God gets mischievous again.
Since he did not want to add 10 to all these numbers and disappoint his followers, he decides to multiply all these numbers by 10.

And once again, surprisingly, nothing changed.
No one noticed the difference.
If one number is less than another, of course, 10 multiplied by the first one is also less than 10 multiplied by the second one. Also, the result you get on dividing one number by another is exactly the same as the result you get on dividing 10 times the first number by 10 times the second number.
Since comparison and division were the only things people did with volume, The Volume God's new mischievous plans had no effect on the world. So does it mean we can freely change the formula for volume of a cuboid to 10 x (length x breadth x height) and the volume of a sphere to 10 x (4/3 x pi x radius cubed) and in general, multiply all formulas for volume by 10 and nothing will ever change? This time, it's actually true.
The exact formula for volume does not mean anything until you specify what unit you are measuring it in.
Multiplying all formulas by 10 is equivalent to defining a new unit and measuring all the volumes in that specific unit. Going back to the question we started with, do we really need all these formulas for volume? Yes, we do, up to multiplication by some constant, or in other words, up to specifying the unit. I leave the similar question about length and area for the reader to ponder.

Why, for example, would anyone ever need the formula for the area of a circle?