You derive the unit of a quantity from its formula, so you do not have to memorise it. Square brackets around a quantity mean 'the unit of'. From distance is speed times time you get metre per second, and from density is mass divided by volume you get kilogram per cubic metre.
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My energy bill shows an amount per cubic metre and an amount per kilowatt-hour. Which is dearer? You cannot compare those two numbers, because they do not belong to the same unit, and that is exactly what this video is about.
In eight minutes you learn one small sign that makes the rest of your physics easier: square brackets around a quantity, to be read as 'the unit of'. With those you can read a formula as if it were about units instead of about numbers, and then you simply derive the unit of speed instead of memorising it. You do the same with density, and then you see straight away what density actually is. You learn why an area is in metres squared and a volume in metres to the third, how you use that to check your own answer, and how to go from kilometres per hour to metres per second without guessing which way to divide.
For upper secondary physics. Halfway there is a question and a silence: pause it there and think for yourself first.
In this video
The full explanation, in writing
The problem
0:00Here is what I pay for gas, and here what I pay for electricity. The one amount goes with a cubic metre, the other with a kilowatt-hour. And now the question everyone asks and nobody can answer: which is dearer? You cannot compare those two numbers, because they do not belong to the same unit.
0:24And that is the problem this whole video is about.
One new sign, worth the whole chapter
0:32There is a shorthand for the phrase 'the unit of'. You put square brackets around the quantity. So instead of 'the unit of length is metre' you write: bracket, l, bracket, is m. Why that is handy you see as soon as you have a formula. Formulas are nothing but a shortened way of writing down a relationship between quantities.
0:57And if the quantities on both sides are equal, then so are their units. In other words: you can work out the units separately, as if it were a sum of its own. And that is what you learn today.
The unit of speed, derived instead of memorised
1:16Take speed. Distance is speed times time. So the unit of distance is the unit of speed times the unit of time. The unit of distance you know: metre. The unit of time as well: second. So metre is the unit of speed times second. And then you turn it round: the unit of speed is metre divided by second. Metre per second.
1:42So you did not look it up and did not memorise it. You derived it. And that works with every formula you meet, even with formulas you have never seen before. One more thing about the writing. Metre per second may be written with a slash, or as metre times second to the minus one.
2:06Those two mean exactly the same. For a unit a negative power is the same as dividing, just as with powers of ten.
What is the unit of density?
2:17Now you. Density is mass divided by volume. Mass you measure in kilograms, volume in cubic metres. What then is the unit of density? The unit of density is the unit of mass divided by the unit of volume. Kilogram divided by cubic metre: kilogram per cubic metre.
2:42And notice what that means. Density is not just a number belonging to a substance. It is literally the mass of one cubic metre of that substance. The unit tells you what the quantity is. That is the second reason to take units seriously. The first was: then you can compare.
3:07The second is: then you understand what you work out.
Powers of units
3:11Area is length times width, and both are in metres. Metre times metre is metre squared. Volume is length times width times height, so metre to the third. That is no accident. The rules you learned with powers of ten hold in exactly the same way for powers of units.
3:32When multiplying you add the exponents, when dividing you subtract them. And that works with the formulas that are less obvious too. Circle, sphere, cylinder: whatever the formula looks like, count how many lengths are multiplied in it. In a circle there are two, radius times radius, because pi is a number without a unit. So square metre.
4:03In a sphere and a cylinder there are three, so metre to the third. If you get something else, you have made a mistake. The unit is your check.
Converting without guessing
4:17Back to the beginning. Two numbers with two different units you compare by first bringing them to the same unit. And you do that not by feel, but with a conversion factor. The classic example: kilometre per hour to metre per second. One kilometre is a thousand metres, and an hour thirty-six hundred seconds.
4:42So one kilometre per hour is a thousand divided by thirty-six hundred metres per second, and that is one divided by three point six. So from kilometre per hour to metre per second you divide by three point six. The other way round you multiply by it.
5:03And if you cannot remember which way: a hundred kilometres per hour is about twenty-eight metres per second. If you get three hundred and sixty, you have multiplied instead of divided. That is how you check it. Safest is to put it all first in the base units: metre, kilogram, second.
5:30Then nothing can go wrong in the rest of your calculation. And that energy bill from before? You can compare it, but not with only what is on the bill. You need something else as well: how much energy is in a cubic metre of gas. That number you have to look up.
5:54And that is an honest answer: sometimes you can only answer a question once you fetch a piece of data. Three things. One: square brackets mean 'the unit of', and with them you read a formula about units instead of about numbers. Two: the unit of a quantity you derive from the formula.
6:18You do not have to memorise it. Three: the unit is your check. If something else comes out than you expect, there is a mistake in your calculation and not in your calculator. Two questions. One: force is mass times acceleration, and acceleration is in metres per second squared.
6:40What is the unit of force, expressed in base units? Two: why are you certain that the answer to a volume calculation cannot be in metres squared? See whether you had the same. One: force is mass times acceleration. Mass is in kilograms, acceleration in metres per second squared.
7:08Together that becomes kilogram metre per second to the minus two. And that unit has been given a name of its own: the newton. And two: volume is length times width times height. Those are three lengths, so metre to the third. Metre squared is only two, and that is an area.
7:32If the number of metres is wrong, the calculation is wrong. From here you never have to look up a unit again. You can derive it.
Frequently asked questions
What do square brackets around a quantity mean?
They are shorthand for 'the unit of'. Instead of 'the unit of length is metre' you write [l] = m. With that you read a formula about units instead of about numbers.
What is the unit of density?
Kilogram per cubic metre. That follows from the formula: density is mass divided by volume, so kilogram divided by cubic metre. The unit tells you straight away what density is, namely the mass of one cubic metre of that substance.
How do you convert kilometres per hour to metres per second?
Divide by 3.6. A kilometre is a thousand metres and an hour is 3600 seconds, so one kilometre per hour is 1000 divided by 3600 metres per second, and that is 1 divided by 3.6. The other way round you multiply by 3.6.
Why is area in square metres and volume in cubic metres?
Because you multiply two lengths and three lengths respectively. With every formula, count how many lengths are multiplied in it: for a circle there are two, because pi is a number without a unit, and for a sphere or a cylinder there are three.
What is the unit of force in base units?
Kilogram metre per second to the minus two. Force is mass times acceleration, mass is in kilograms and acceleration in metres per second squared. That unit has been given a name of its own: the newton.
From here you never have to look up a unit again. You can derive it.
Physics with Thomas makes physics explainers for upper secondary school. No words on screen, so what you see works in any language.
In this series
Quantities and unitswhy 20,000 is not more than 10,0007:30
Ratios and percentagesone table for both3:15
Scientific notationpowers of ten and order of magnitude5:22
Calculating with powers of tenthe four rules3:18
Pythagoras and trigonometrywhich side with which angle3:19
Deriving units from a formulaconverting to base units7:47
Significant figureshow many figures do you write6:24
Line of best fitthe line touches no point5:07
Relationships and the constantbraking twice as fast5:20
Linearising a graphyour measurements never change4:41
About me
My name is Thomas Schuurmans, and I have a passion for physics and for understanding why things work the way they do. What I want is to get young people asking “why is that?” and “how does that work?” more often, and to hand them the tools of physics in a way that is simple and that you can see.
I graduated in applied physics at Delft University of Technology in 2003. After that I spent more than twenty years outside education: first at TNO, the Dutch applied research institute, then at a design agency, and eventually founding Proportion Global, through which I work on innovation questions in Africa, Latin America and South Asia. That work resembles physics more than you would expect: don't start from a solution, first understand what is going on, try something, be wrong, and look again.
Since 2026 I have been teaching physics to both lower and upper secondary classes, and I started a master's at the University of Amsterdam for my full teaching qualification. That is where I learned that a secondary school pupil's real attention for new material lasts about seven minutes. And my own weakness happens to be telling too many side stories.
That is where the idea came from: videos that explain one topic sharply and visually, inside those seven minutes. I make them for my own pupils. Then I publish them, because good explanation should be within reach of anyone who needs it, wherever you live and whatever language you think and speak in.