### Video Transcript

In the metric system, there are ten millimetres in a centimetre;
there are a hundred centimetres in a metre and a thousand metres in
a kilometre. Some questions give you dimensions using one of these units and ask you to give
your answer using another. This is reasonably straightforward when you’re dealing with
lengths; but when you start converting areas and volumes, the differences can be quite
surprising.

Let’s have a look at a couple of questions. So firstly, let’s consider
centimetres and metres: one metre is the same length as a hundred
centimetres, so for every metre we’ve got a hundred centimetres. Now question one says write seven metres in
centimetres. So that makes seven lots of one hundred
centimetres, which is seven hundred centimetres.

Now let’s look at our second question: write three hundred and
fifty centimetres in metres. Well that’s three lots of a hundred centimetres, so
that’s three individual metres and then fifty centimetres. And
fifty centimetres is half of a metre, so nought point five metres. So
three hundred and fifty centimetre is three and a half metres in
total.

So looking again at that process, in the first question, converting from
metres to centimetres, every one metre was a hundred centimetres so
we had to multiply the seven metres by a hundred to get seven
hundred centimetres. In the second question, every one hundred
centimetre was only one metre, so we had to divide by one hundred in
order to find the answer in metres. So rather than doing all the diagrams, I could’ve just said seven times
a hundred is seven hundred centimetres or three hundred and fifty divided by a
hundred is three point five metres.

Now let’s think of a square. Now this square is one metre across
and it’s one metre up, and the area therefore is one times one,
which is one square metre, but what would that area be in square centimetres? Well one metre is a hundred centimetres. So instead
of doing the calculation in terms of one metre across and one metre
up, we’re going a hundred centimetres across and a hundred
centimetres up.

So our area is a hundred times a hundred, which is gonna be a hundred then times by ten and
times by ten again. So it’s gonna be a one with four
zeros after it; that’s ten thousand square centimetres. So although the individual dimensions are being multiplied by a
hundred to convert metres into centimetres, the answer is being multiplied by a
hundred times a hundred, is being multiplied by ten thousand, in order
to give us our answer in square centimetres.

Now let’s look at a cube, which is one metre by one
metre by one metre. In this case, the volume is again one times one times
one, so it’s one cubic metre. But what would that be in cubic
centimetres? Well each of our one metres are now one
hundred centimetres, so the volume calculation in cubic centimetres is a hundred times a
hundred times a hundred.

So that’s a hundred times ten times ten times ten times ten;
that’s a one with six noughts after it, a million cubic centimetres. So again, our linear dimensions were multiplied by a hundred, so
there are a hundred times more centimetres in that length than there are metres.
But that means that the volume calculation had to be multiplied by a million in
order to get the right answer in cubic centimetres.

Now if you think about this, if you visualise this, if we had little
one-centimetre cubes, we would need a hundred of them laid end to
end to make one row across the front of this
one-metre-by-one-metre-by-one-metre box. And if I
then had a hundred of those rows going backwards across here, that would be
a hundred times a hundred of those one-centimetre cubes. And then
I’d need a hundred layers just the same in order to fill up the-the
one-metre cube there, so a hundred times a hundred times a
hundred; I would need a million of these little cubes in order to fill out
that one-metre cubed box.

So just for conversion between metres and centimetres, if we consider areas
and volumes, we’ve not only got one number to remember, a hundred
centimetres in a metre, we’ve got three numbers to remember: the values for converting areas
and volumes as well. And if we think about all of the different units of length in the metric
system, it would take a great deal of effort indeed to remember all the conversion factors for
lengths and areas and volumes between each and every one of these.

To be fair, most exam questions are only going to ask you to convert between
metres, centimetres, and millimetres or even kilometres, but even that will be a lot to
remember: twelve factors for lengths, twelve more for areas, and
twelve more for volumes. So the top tip then is if we just aim to learn the basic length
relationships, we can easily work out the rest when we need them.

So for example, if we know the number of kilometres, we just need to multiply
that by a thousand to find out how many metres we’ve got; if we know the number
of metres, we just times it by a hundred to work out the number of centimetres;
and if we know the number of centimetres, we multiply it by ten to find out how
many millimetres. And the reverse, we just have to divide by those numbers to work our way back
along that line. Obviously every centimetre’s got ten millimetre in it. So if we
know the number of millimetres, we just divide that by ten and that will tell us
the number of centimetres and so on.

Okay, let’s try a question: convert three point seven five
square metres into square centimetres. Now a nice easy method for these area questions is to just draw a rectangle,
and we want to draw a rectangle that’s got an area of three point seven five
square metres. Well let’s make one of the sides one metre, so the other one must
be three point seven five
because one times three point seven five is three point seven five
square metres.

Now we can convert each of the individual lengths into centimetres;
one metre is a hundred centimetres; three point seven five
times a hundred is three hundred and seventy-five centimetres. So to work out our area in square centimetres, it’s simply one hundred
times three hundred and seventy-five. So that’s like multiplying by ten
twice. So that’s a three seven five with two zeros on the
end: thirty-seven thousand five hundred square centimetres.

Let’s look at another question then: convert eleven point six
square metres into square millimetres. Again, let’s start off by drawing a rectangle that’s got a height of
one metre. So the length would need to be eleven point six metres because
eleven point six times one is eleven point six to give us our eleven
point six square metres.

Now if I multiply each of those lengths by a hundred because I
know that there’re a hundred centimetres in a metre, I’ve got a
hundred centimetres by one thousand one hundred and sixty centimetres.
But remember, I want my answer in square millimetres so I’ve gotta do another conversion. Because each centimetre is made up of ten millimetres, I’ve got
to multiply those numbers by ten again in order to work out how many millimetres
they had, and now I can work out the area.

Just multiply those two numbers together so that’s one one six o o
times ten
three times, so that’s another three zeros on the end; That’s eleven million six hundred thousand square millimetres.

Let’s have another one then. Convert two point nine square
kilometres into square metres. So first of all, we’re gonna draw our rectangle. First let’s make the height one kilometre. Then think about how
long would that rectangle need to be to make an area of two point nine square
kilometres. Well it would need to be two point nine kilometres.

Now we can convert them into metres. Well a kilometre’s got a
thousand metres in it. So multiplying each of those numbers by a thousand gives us
a thousand metres by two thousand nine hundred metres.

So the area, we’re gonna multiply those two figures together, a
thousand times two thousand nine hundred is two thousand nine hundred with
three zeros added to the end, which makes two million nine hundred thousand square metres.

Okay let’s remove some of the clues and do a slightly harder question. We’re
now gonna go for volume, so convert fifteen point two cubic metres into cubic
centimetres. Well this time we’re not gonna draw a rectangle, we’re gonna draw a cuboid. And to work out the volume of a cuboid, remember, it’s the area of the end
times the length. So if we make that area of that end of that cuboid equal to one
square metre, then the length of the cuboid is gonna have to be fifteen point
two. So in order to do that, I’m gonna make the width one and the height
one.

So the area of the end of that cuboid is one square metre. And the length is gonna be fifteen point two metres. So we’ve
created a cuboid with this volume and now we’ve got to convert it into cubic centimetres. And because there are a hundred centimetres in a metre, we have
to multiply all of those measurements, those length measurements, by a hundred.
So we’ve got a hundred centimetres by a hundred centimetres by
fifteen hundred and twenty or one thousand five hundred and twenty
centimetres.

And a volume calculation is gonna be a hundred times a hundred times
one thousand five hundred and twenty. So what I’m gonna do is I’m multiplying by
ten and multiplying by ten and multiplying by ten and
multiplying by ten; I’m gonna end up with one five two zero with
four more zeros on the end of that. So that’s fifteen million two hundred thousand cubic
centimetres.

Now top tip to make your numbers more readable in an exam so that the
examiner can count things up nice and easily is to leave spaces in your numbers. So where we’d
usually start is from the-the right-hand end, the one’s column here, okay? The
first three will group together, then we’ll leave a little gap here, then we’ll group The Next
three together, then we’ll leave a little gap here, and so on. So leaving little spaces like that every three digits, starting
from the right moving left, it’s easier to see that that’s fifteen million two hundred
thousand.

Now we’re gonna do a monster example and convert cubic kilometres into cubic
millimetres. And if you think how many cubic millimetres there are in a cubic kilometre, the
conversion factor’s gonna be phenomenal. But let’s do this from first principles then. This is a pretty big cuboid by any standards, so we’re gonna do the standard
one kilometre high, one kilometre deep. So that means the length
would have to be one point eight kilometres in order to give us a volume of
one point eight kilometres cubed.

So first I know that there are a thousand metres in a kilometre,
so I’m gonna multiply those numbers by a thousand to convert them into metres. Then I know that there are a hundred centimetres in a metre, so
I’m gonna multiply all those numbers by a hundred.

Finally, I know that there are ten millimetres in a centimetre,
so every one of those centimetres counts for ten millimetres. So multiply the
numbers by ten and my cuboid then is a million millimetres high,
a million millimetres deep, and one million eight hundred thousand
millimetres long.

So multiplying all of those together, I’m gonna get the volume in cubic
millimetres. So that’s gonna be basically one million eight hundred thousand,
then I’m gonna add another six noughts cause I’m multiplying by ten
six times here and another six noughts cause I’m multiplying by
ten
six times here.

So that’s this number here, and I’m gonna do the exercise of sort of putting
in my little spaces every three as I go through to make this a little bit clearer. So there’s my number, very big. For those who are interested, that’s
one point eight quintillion cubic millimetres.

Now so far we’d be making all our numbers bigger, but converting in the other
direction is a little bit trickier. You’ve gotta be very careful about decimal places, but the
same basic method makes life so much easier than trying to remember all the area and volume
scale factors individually. So we wanna convert twenty-five square millimetres
into square centimetres.

So we draw our rectangle because it’s areas, and we start off trying to
create the original number. So it’s gonna be — the units are millimetres so our units here are
gonna be millimetres, and we’re gonna have a height of one millimetre because
that’s what we always do. And to make that rectangle have an area of twenty-five
square millimetres, the length would need to be twenty-five.

So to convert the length from one millimetre is we’ve gotta
divide that by ten. Remember, there’re ten millimetres in a
centimetre, so one millimetre is a tenth of a centimetre or
nought point one centimetres. Twenty-five millimetres, divide that
by ten, is two point five centimetres.

So you will see then this is slightly more difficult. It’s the same basic
process. But because we’re dividing by ten, we’re ending up with decimal places
everywhere, which just makes it a little bit more tricky to deal with. And to work out the area we’re taking a two point five length
and we’re multiplying this by null point one. Now remember null point
one is basically we’ve had to divide by ten in order to get that. So
we’re going to be dividing by ten, and that gives us null point two five square centimetres.

Next time we want to convert fifty-four square centimetres into
metres squared. So using our original units of centimetres, we construct the rectangle; it’s
always gonna be one centimetre high. How long would that rectangle need to be to
create an area of fifty-four square centimetres? Well it would need to be
fifty-four centimetres long.

Now we’re converting the answer into square metres, so we’ve got to get these
length units into metres. And there are a hundred centimetres in a metre, so to
convert centimetres into metres we need to divide by a hundred. And that gives us a height of one divided by a hundred is null point
null one; fifty-four divided by a hundred is null point five four
metres.

So we’re gonna have to multiply those two together to get our area. Now
remember that this multiplying by null point null one is the same as dividing by
a hundred, so I can just do this little trick of moving the decimal point
two places back. So remember the decimal point was in between the zero and
five here; I’ve moved that back two places and then filled in all
the zeros around it. I’ve got zero point zero zero five four square metres.

And lastly, let’s do another monster example in the other direction this
time: convert null point three cubic metres into cubic kilometres.

Well this is a volume question, cubic metres, cubic kilometres, so we’re
going to draw a cuboid rather than a rectangle. Now we want our cuboid to have a volume of
null point three cubic metres. I said, remember, we always do a height of
one and a depth of one based on the units that we were given. So
that’s metres in this case, so that’s one metre by one metre. Now to
create a volume of null point three cubic metres, so let’s do one times
one times something to create null point three. That’s got to be
null point three metres. So yes I haven’t exactly drawn it accurately; that
null point three should be a lot shorter. But we’re not doing scale drawing
here, we’re just doing a little sketch to help us to do our calculations more easily.

So the next stage, we’ve got metres; we’ve got to convert them into
kilometres. Now there are a thousand metres in a kilometre. So if we know the number of metres, there will be a thousand
times fewer kilometres because, obviously a thousand metres will be
one kilometre, so we divide all of those numbers by a thousand. So that’d be all; we divided all those numbers by a thousand.

So little top tip here, we’re doing one and we’re trying to
divide it by a thousand; so write the number out with a decimal point so
one point zero. And we’re dividing by ten
three times cause we’re dividing by ten; we’re dividing by another
ten; we’re dividing by another ten; a thousand is ten times
ten times ten, so we’re dividing by ten
three times here. So if I divide by ten once, my decimal point would
move to here; twice, it’s gonna move to here; third time, it’s gonna move to here. So this is
where my decimal point has gone; let’s cross that one out. And I would have a
zero in here under this little arch, a zero under here under this
little arch, and then I need a zero in front of that. So zero point zero
zero one is one divided by a thousand.

And that kinda makes sense because the first digit here is the tenths column,
the second digit here is the hundredths, column and the third digit here is the thousands
column. So one divided by a thousand is one
one-thousandths.

So the volume is simply those three dimensions multiplied together. But don’t
forget, we cunningly set up this example, in fact all of these volume examples. This technique
is based around the fact that because we had one metre and we had
one metre, here we’re dividing by a thousand and here we’re
dividing by a thousand. So multiplying by nought point nought nought
one is the same as dividing by a thousand.

So dividing by a thousand, we’re gonna move our decimal point to
the left three places; dividing by a thousand we’re gonna move our
decimal point to the left another three places. And there we have it: nought point nought nought nought nought nought
nought nought nought nought three cubic kilometres.