Lesson Video: Measuring Lengths Physics

In this video, we will learn how to correctly use rulers to measure lengths.

17:31

Video Transcript

In this video, we will learn about how to measure lengths correctly and accurately using a ruler. Now to see why this is important, let’s first imagine the following scenario. Let’s say that we are in a carrot farm. Here it is, look, Nagwa carrot farm. And look, here we are.

Now let’s say that, on this farm, we pride ourselves on growing extremely long carrots. And what we want to do is to invite over an official from the Strongbow Book of Records. We want them to measure our longest carrot because we reckon we’ve got a world-record-length carrot.

However, we need to be absolutely 100 percent sure because the current world record for carrot length is held by our arch nemesis farmer Gary. Now we don’t really wanna go through the embarrassment of inviting over the Strongbow Book of Records official only to find that our longest carrot isn’t really the longest carrot. If that were to happen, then farmer Gary would not let us forget about it.

So what we need to do is to make sure that we really do have the world-record-breaking carrot. To do that, we need to know how to measure the length of our carrot with a ruler. So how do we go about doing this?

Well, before we learn how to measure the length of a carrot, let’s simplify things a little bit and learn how to measure the length of a straight line. The same principles will apply to the carrot. But measuring a straight line is much easier. So how are we’re gonna go about doing this?

Well, if we wanna measure the length of the straight line with our trusty ruler, then we need to remember three simple steps. Step number one: Line up the ruler with the straight line so that they are both parallel to each other. Specifically, the edge of the ruler that we use to measure with is meant to be parallel to the straight line.

Number two: Move the ruler or the line if the line can move underneath the ruler so that the zero marking on the ruler — that’s this marking here — lines up with one end of the line. So now in this case, what we’ll imagine is that the line is drawn on a piece of paper and we can move the paper underneath the ruler. This way, we can move the paper until the end of the line is aligned with the zero marking on the ruler.

Now in a situation like this, we need to be very careful. When we make sure that the end of the line is aligned with the zero marking, we need to be looking at that part of the ruler and the line from directly above. To understand why, let’s draw a diagram from a slightly different viewpoint.

Let’s imagine that we move around so that we’re now looking at the diagram that we’ve just drawn from this point here. Let’s say this is a new eyeball position. And we can see all of what’s going on with the ruler measurement. Well, in that case, what we would see is the straight line and the ruler being brought near to it, where the ruler has some small thickness. It’s not very thick, but it does have some thickness. But anyway, that’s just to set the scene.

So now let’s imagine that the person doing the measuring comes over to the ruler and is trying to align the zero mark, which let’s say is over here on the ruler, which we can label in pink to make more clear, with this end of the straight line.

Now if the person is not looking from directly above, then they would think that, based on their line of sight, the zero mark is actually well to the left, as they see it, of the straight line, or the way that we’re looking at it in our diagram well to the right. However, in reality, the zero mark is actually this side of the end of the straight line. And so simply by not looking at it from directly above, the person doing the measuring has failed to realize that the ruler is not quite in the right position.

This is actually problematic because let’s say somebody before this person had come along and set the ruler correctly so that the zero mark was aligned with the end of the straight line. This person would now see the zero mark to still be too far left, as they see it, of the straight line. This is of course an exaggeration, but it is a genuine problem. We need to make sure that that doesn’t happen. We need to make sure that when we’re doing this alignment, we’re looking from directly above the zero mark and the end of the straight line. So that’s step number two, which means that we can move on to step number three for measuring our straight line’s length.

Step number three is to take the reading at the other end of the line. In other words, we read off where this end of the line corresponds with the reading on the ruler. And of course, we have to do the same thing again, making sure that we’re directly above this point so that we don’t get any measurement error that we saw earlier, which is known as a parallax error.

So in this case, we can say that the length of the straight line is 11 centimeters because, in going from this end to this end, the line ends up being one centimeter long, two centimeters, three four five six seven eight nine 10 and 11 centimeters long. So if we wanna measure the length of a straight line with a ruler, we need to remember these three steps. And of course, the same applies if we’re trying to measure the length of a carrot.

Speaking of carrots by the way, it turns out that the real-life longest carrot is over six meters in length. That’s longer than any ruler we’re likely to use. So to measure its length, we’d probably have to use a tape measure or multiple rulers stacked next to each other. But we’ll see how to do that later on. For now though, let’s look at a few examples.

Scarlett uses a centimeter ruler to measure the length of a straight line as shown in the diagram. She determines that the length of the line is 10.4 centimeters. Which of the following statements explains why this answer is incorrect? A) The ruler is not parallel to the line. Thus, the line is actually longer than 10.4 centimeters. B) The ruler is not parallel to the line. Thus, the line is actually shorter than 10.4 centimeters. C) Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 11 centimeters. D) The maximum resolution of the ruler is one centimeter. Thus, the length of the line should be recorded as 10 centimeters.

Okay, so in this question, Scarlett has measured the length of line to be 10.4 centimeters because she’s taken the reading at this point on the ruler. And we’ve been asked to state why this is incorrect. So immediately, we can see that options C and D are not the right answer.

Let’s consider option D first. This option says that the maximum resolution of the ruler is one centimeter. Well, already that’s untrue. Of course, we do have markings every centimeter along the length of the ruler. But we also have smaller markings every millimeter along the ruler. And so the maximum resolution of the ruler is not one centimeter. And hence, option D is not the correct answer to our question.

Looking at option C then, this one says that measurements using a ruler should always be rounded up. Well, that is actually a very bad idea because we introduce a systematic error if we do this. Any length that’s not perfectly on a centimeter mark will always be rounded up, which means we’ll systematically measure objects to be longer than they actually are. And this is not a good thing. Therefore, option C is not the correct answer to our question either.

That means that it’s out of one of options A and B. Now both options start off with the same thing. They both say that the ruler is not parallel to the line. The ruler is not parallel to the line. And they both state that this is the reason why the measurement is incorrect. And if we think back to our rules from earlier, rule number one when measuring the length of a line with a ruler was to make sure that the measuring edge of the ruler and the straight line are parallel to each other. And in this case, they’re actually not.

We can see that the edge of the ruler with which we measure is pointing in this direction. However, the line is pointing in this direction. So these two are not parallel. But then what is the consequence of this?

That is where options A and B differ from each other. Option A says that because the ruler and the line are not parallel, this means that the line is actually longer than 10.4 centimeters. That’s longer than what Scarlett measured it to be, whereas option B says that because the ruler and the line are not parallel, that means that the line is actually shorter than 10.4 centimeters. So which is it?

Well, we can see first of all that although the line and the ruler are not parallel, Scarlett has done step two correctly. This end of the line is perfectly aligned with the zero mark. So what we can do here is to imagine that we rotate the straight line about the left-hand end. And we do this until the line is parallel to the ruler. So here’s our line X part way for being rotated. And here is our line once it’s parallel to the ruler.

In this situation, we can see that the reading on the ruler is now going to be larger than even 11 centimeters. This means that the line is actually longer than what Scarlett measured it to be, which was 10.4 centimeters. And that corresponds with option A. So option B is not the answer to our question either. So we’ve got our correct answer now.

The reason that Scarlett’s measurement of the length of the line X is incorrect is because the ruler is not parallel to the line. Thus, the line is actually longer than 10.4 centimeters.

Okay, so in this example, we saw what happens when step one of our measurement process is not done correctly. Let’s see what happens when we make a mistake on another one of our steps.

Sophia uses a centimeter ruler to measure the length of a straight line as shown in the diagram. She determines that the length of the line is 9.4 centimeters. Which of the following statements explains why the answer is incorrect? A) The zero marker of the ruler was not positioned at the start of the line. Thus, the line is actually shorter than 9.4 centimeters. B) The zero marker of the ruler was not positioned at the start of the line. Thus, the line is actually longer than 9.4 centimeters. C) The ruler is not parallel to the line. D) The maximum resolution of the ruler is one centimeter. Thus, the length of the line should be recorded as nine centimeters. E) Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 10 centimeters.

Okay, so first things first, here’s the line that we’re trying to measure. We can see that the line is pointing in this direction, and so is the edge of the ruler that we use to make measurements. They’re both aligned. And therefore, the line and the ruler are in fact parallel. This means that option C is immediately ruled out because they say that the ruler is not parallel to the line.

So Sophia has done this correctly. She has aligned the ruler with the straight line. Now let’s consider options D and E before we do anything else. Looking at option D, this says that the maximum resolution of the ruler is one centimeter. Well, this is clearly not true. Of course, there are markings every centimeter, but there are also smaller markings every millimeter. And so the maximum resolution of the ruler is not one centimeter. It’s actually one millimeter. So option D is not our answer either.

Moving on to option E, this one says that measurements using a ruler should always be rounded up. Well, that is definitely bad practice. We should not be doing this, because if we’re rounding everything up, then unless some length that we’re measuring is perfectly on a centimeter mark, we would always end up rounding it up. And so we would always measure it to be longer than it actually is.

The idea with rounding though is that, on average, half of the time we would round up and half of the time we would round down. This way, any errors in the rounding up would be canceled out by the errors in the rounding down. That’s if we’re measuring different lengths of course. But if we’re always rounding up, then we’re introducing a systematic error. And so option E is out of the question as well.

So let’s now look at options A and B. They start off very similarly. Both of them say that the zero marker of the ruler was not positioned at the start of the line. Option B says exactly the same thing. And that, in fact, is true. Here’s the zero marker of the line, and here is the start of the line. So this is going to result in an error in the measurement. But what is the consequence of this?

Well, option A says that the line is actually shorter than 9.4 centimeters. That’s shorter than what Sophia measured it to be, which happens to be a reading here at 9.4 centimeters on the ruler, whereas option B says that the line is actually longer than 9.4 centimeters. So which is it?

Well, to answer this question, we need to imagine that we correctly placed a ruler next to the line. We need to ensure that the zero mark — that’s this mark over here — meets up with the start of the line. To do this, we would either have to move the ruler left or move the straight line to the right. Either way, we would have to do this until this end of the line moved to the zero mark. And we can imagine this as just moving the straight line to the right.

But then in that situation, this end of the line moves to the right as well. And so once we’ve aligned the line correctly, the actual reading is going to be larger than 9.4 centimeters. It’s gonna be somewhere closer to 10 centimeters. And hence, the line is actually longer than 9.4 centimeters. That is what option B says, and so option A is incorrect.

Hence, we found our answer to the question. The length of the line as measured by Sophia is incorrect because the zero marker of the ruler was not positioned at the start of the line and thus the line is actually longer than 9.4 centimeters.

Okay, now earlier, we mentioned that if we’ve got a really long carrot, we might have to measure it with multiple rulers stuck together. Let’s have a look at a question that shows us how we’d go about doing that, except where we’re measuring a straight line again rather than a carrot. But I hope that’s okay.

Liam uses two centimeter rulers to measure the length of a straight line as shown in the diagram. He determines that the length of the line is 19.2 centimeters. Which of the following statements explains why this answer is incorrect? A) The rulers are not parallel to the line. B) The two rulers have not been placed end-to-end. C) The maximum resolution of the ruler is one centimeter. Thus, the length of the line should be recorded as 19 centimeters. D) The second ruler has been placed the wrong way around. E) Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 20 centimeters.

Okay, so in this situation, we’ve got a line here. And Liam is trying to measure its length. He realizes that one ruler is not enough and that the line is too long for the ruler. So he has to use a second ruler. Let’s first think about the things that Liam has done correctly.

Well, we can see first of all that the zero marking on the first ruler is aligned with the start of the straight line. So that much has been done well. Secondly, we can also see that both rulers, or more specifically the edges used to do the measuring on the rulers, are parallel to the straight line. They’re all aligned in the same direction.

And coincidentally, that rules out option A. Option A says that the two rulers are not parallel to the line, where in reality they are. So this much Liam has done well. So where has it gone all wrong?

Well, let’s consider option B first of all. This option says that the two rulers have not been placed end-to-end. Well, it’s true that the two rulers have not been placed end-to-end, but that is actually a good thing. Let’s say that the two rulers had been placed end-to-end and this was the line we were trying to measure the length of.

Well then, there would be a whole chunk of this ruler and a whole chunk of this ruler, which wouldn’t be measuring anything because there are no markings on the rulers past 12 centimeters. This means that these bits of plastic on the ends of the ruler past the final marking could be arbitrarily long or short. And they wouldn’t be measuring anything. In effect, all what we’d be measuring is this length ending here and this length starting here. This whole bit would’ve been completely ignored.

So although the two rulers have not been placed end-to-end, this is a good thing. And so option B is not the correct answer to our question. Option C then, this says that the maximum resolution of the ruler is one centimeter. Well, let’s just stop there. We’ve been told that we’ve got two centimeter rulers. This means that the big markings on the rulers are every one centimeter. But we can also see that there are little markings every millimeter, every tenth of a centimeter. And so the maximum resolution of the ruler is not one centimeter. So immediately, we can cancel out this option.

Option D then, the second ruler has been placed the wrong way around. Well, that’s true. The second ruler has been placed the wrong way around. We can see that this ruler has its numbers the right way up, whereas the second ruler has its numbers upside down. So this might be the source of the problem. And in fact, that is the reason why. We’ll come back to this in a second.

However, let’s quickly make sure that option E is incorrect. Option E says that measurements using a ruler should always be rounded up. Well, why would we want to do that? Why would we want to introduce a systematic error where we always round up unless the length that we’re measuring is perfectly on a centimeter mark? This would mean that anything that’s 19.2 centimeters would be recorded as 20 centimeters and anything that’s 19.8 centimeters would also be recorded as 20 centimeters. That doesn’t make sense. So option E is wrong.

So let’s come back to option D then and work out why that’s the correct answer to our question. We can see that Liam has measured the length of the line up until this point very correctly. Liam has measured zero centimeters, one centimeters, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

But then because he’s placed the second ruler upside down, he thinks that the length of the line from here to here is seven and a bit centimeters. Specifically, that’s the second mark nearest to the seven. So he thinks that part of the line is 7.2 centimeters long.

However, what he hasn’t realized is, by placing the ruler upside down, his 12-centimeter mark now becomes the zero-centimeter mark. And that way, he can basically measure this length — that’s 12 centimeters — and this length separately. Then he can add them together to give the total length of the line.

But anyway, so placing the ruler upside down means that this becomes the zero-centimeter mark. That means that the 11-centimeter mark now becomes one, 10 becomes two, nine becomes three, eight becomes four, and seven becomes five centimeters. And so this part of the line is not 7.2 centimeters long. It’s zero, one, two, three, four, five, 5.2 centimeters long. Hence, the total length of the line is 12 centimeters from the first ruler plus 5.2 centimeters from the second ruler. And that is equal to 17.2 centimeters, not 19.2.

Now Liam could’ve equally made the second part of the measurement by placing the second ruler the correct way around — that’s the right way up — and overlapping the zero mark on the second ruler with the 12-centimeter mark on the first ruler. This way, the first part of the line already measured to be 12 centimeters can now be ignored. And anything ahead of this is an additional centimeter on top of the 12 centimeters. And two centimeters ahead means we’ve got the 12 centimeters from earlier plus two centimeters here, and so on and so forth, which in this case ends up being 5.2 centimeters more than the original 12 centimeters. So in reality, the second ruler has been placed the wrong way around. And that is the answer as to why Liam got an incorrect measurement.

Okay, so let’s summarize what we’ve learnt in this lesson. So when we’re trying to measure the length of a straight line with a ruler, we need to remember three simple steps. Number one: We need to make sure that the line and the ruler are parallel. Once we’ve done that, step number two is to align the zero marker with the start of the line. So that’s this zero marker on the ruler aligned with the start of the line.

And it’s important to remember by the way that when we do this, we need to be looking at the ruler from directly above. This way, we avoid any parallax issues. And finally, when we’ve done these first two steps, we take a reading at the other end of the line. So in this case, we see that this end of the line is seven centimeters away from the zero marker. Therefore, the line is seven centimeters long.

And once again, we need to take this reading whilst looking from directly above the seven centimeters’ mark. Oh and by the way, we got the world record. We had the longest carrot. Take that farmer Gary.

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