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Find how to reliably convert lengths, areas, and volumes given in one metric unit to another metric unit without remembering all the scale factors.
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.
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