This isn’t necessary until the end, buuuuut…..
Introduction
When we talk about how big something is, often times we use similar looking units but are in fact quite different. For example m, m2 , and m3 are each very different things.
Length
An Olympic sprint can be 400 m long. This is a measure of distance and the length of travel is only what matters.
Area
However, when you are looking for an apartment and you ask the landlord how big the master bedroom is, an answer of “5 meters” would be confusing and incredibly incomplete as this is only the length. You might consider digging a little further before signing any contracts. Consider two apartments that are equal in all ways (including price) other than their master bedrooms.
Master Bedroom A is 5 meters long but only 1 meters wide would give you some space management issues….for most adults, they would only be able to lie down in the room lengthwise. This is awkward.
Master Bedroom B is 5 meters by 8 meters. This will be more than enough space for the normal college freshman’s full size bed, desk, television, and more. The size of the apartment is best determined here by taking in how much area is taken up here.
Remember, for a rectangle: 
, area equals length times width.
Master Bedrooms in each apartment:
Volume
Consider: You own land in Cambodia and want to build a massive luxury hotel. You have many things to consider. But, like, even though this is totally obvi, the biggest decision you have to make is…….
How big is your pool going to be????
(…duh.)
The above picture is of the pool at the Sofitel Hotel in Siem Reap, Cambodia. This entire place is like ridiculously posh. ($150 a night? Seriously??!? Seriously.) Important: if you learn nothing else from this section, learn this: THE DESSERTS ARE AMAZING!
But just how massive is this sprawling lagoon of artificial pool-y perfection you’ve brought into the world? We could talk about square feet, and for some purposes, that may be sufficient.
However, for you, the owner of this soon-to-be watery awesome thing, you have a lot more to consider than just how big it looks from above. You need to consider how much dirt needs to be moved and how much water you will need to fill it. And how much water it takes to fill significantly impacts the effort it takes to maintain your small little piece of paradise. Each of these things come with significant costs that are largely affected how deep the pool is.
To do this, we have to add another dimension to area and we get Volume.
For a basic rectangular prism, volume is calculated as: 
, volume equals length times width times height.
Varying each of these dimensions (length, width, height) changes the volume.
To see how changing dimensions changes length, area, and volume….
Notice, as you move the points around, you can get area which is measured in square units. Notice what are also inside the rectangle……THE SAME NUMBER OF SQUARE UNITS!!!!
Similarly, volume is a measurement of cubes, hence m3.
That concludes this little exploration of dimensions. On the Desmos Demo, make sure to fiddle with the two slider settings on the left….they’re kinda fun…..especially the slope.
