0.9cmx0.9cm = 0.81cm²
9mmx9mm = 81mm²
Do these two numbers equate?
In: 0
The tricky thing to remember is that the prefixes are squared to if you square the unit.
“kilo-” might mean a thousand for every other unit, but for square meter it means a thousand squared, that is a square kilometer is a million square meters.
“centi-” might normally mean one hundredths of a unit, but for squared units it means one thousandths.
There are ten millimeters in centimeter, but one hundred square millimeters in square centimeters.
IF you go with cubed an volume there are a million cubic centimeters to a cubic meter.
It is easiest if you have all your numbers in the same unit with the same prefix because then you can just ignore them. If you do need to convert the result afterwards just remember that different conversion rues apply for squared and cubed units than for regular ones.
It might help to think of the k for kilo and c for centi and m for milli as being more constants with fixed values of 1000 , 0.01 and 0.001 and imagine brackets around the unit names.
1 km^2 is really 1 (k m)^2 = 1 (1000 m)^2 = 1 x 1000^2 x m^2 = 1,000,000 m^2
So yes 0.81cm^2 = 81 mm^2
When you multiply 0.9 cm with 0.9 cm you multiply 0.9 x 0.01 x m x 0.9 x 0.01 x m and get 0.81 x 0.0001 x m² which is 0.81 cm^2 or 81 mm²
Bonus fact this behavior of square and cubed units means that normally with metric unit you have a new bigger or smaller unit every thousand units, but every million for squared and every thousand million for cubed units. (you also have rarer prefixes for 10, 100, a tenth and hundredth, that become hundred, ten-thousand and a hundredth , a tenthosuandth for square and a thousand and a million and a thousandth and millionth for cube) This makes the gaps much bigger than normal.
A centimeter normally is only ten time larger than a millimeter but a cubic-centimeter is a thousand times larger than a cubic millimeter.
This has led to the use of some helper units, if you want to measure some volume that is small than a cubic meter but bigger than a cubic millimeter you have a problem. There are a million cubic centimeters in a cubic meters and perople don’t like to measure their groceries in thousands or tens or hundreds of thousands of something.
This is where a liter comes in handy. Technically a liter is just a cubic decimeter. So one tenth of a meter cubed. Or a thousandth of a cubic meter. Even better the new helper unit of the liter takes the same prefixes as other units without any of the confusion of cubing. A millilitre is just a thousandth of a liter no cubing or squaring involved.
Liter also take the rare pefix of hecto- for hundred. giving you hetcoliter as in 100 liter or a tenth of a cubic meter
For land we have similar units in the form of are and hectare. An are is a square decameter or an area 10 meters by 10 meters. a hectare is a hundred of those or 10,000 m^2
These pseudo metric units of area and volume help as they don’t have the issue of prefixes not meaning what they normally mean because they are implied to be squared and cubed too and they help fill out the gaps that come into existence because the prefixes get pulled further apart by squaring and cubing.
This all might seem terrible confusing but it is just moving the decimal point around according to consistent rules and can be done in your head and still much easier than for example trying to figure how to translate inches of rain per square mile into gallons.
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