Lesson Video: Drawing and Measuring Bearings Mathematics

Video Transcript

In this video, we’ll learn how to draw and measure bearings. We begin, though, by actually discussing what a bearing is and why they’re seemingly so important in the world of transport.

Imagine you’re on a ship in the middle of the ocean and you’re looking to navigate to an island. Unlike if you were in a car, you cannot simply look for points of reference, like an oak tree or a supermarket or even road names and navigate using those. So, instead, we use something called a bearing. A bearing is a fancy way of measuring an angle. But, of course, we need to ensure that every ship captain and indeed every pilot of a plane measures these angles in the same way. And so, we apply some rules.

The rules for measuring and drawing bearings are as follows. First, we measure from north or the north line. Now, this means that if there isn’t a north line drawn on the diagram, we do need to add one. Next, we measure from our north line in a clockwise direction. Finally, we tend to use three digits to describe our bearings. So, 43 degrees as a bearing would be zero, four, three. Now, obviously, this is a great idea in navigation because if you’re communicating with someone else, you’ll know that you haven’t missed any digits if you use three figures all the time.

Depending on where you are in the world, though, when answering questions that don’t involve transport, you might find that 43 degrees would simply be written as 43 degrees. In some places, this would even be written using compass points such as north 43 west, and so on. The terminology also takes a little bit of getting used to. For example, if we’re measuring a bearing of 𝐵 from 𝐴, we’re going to measure the bearing at 𝐴. And so in the case of our diagram, we start at the north line, measure the angle in a clockwise direction that this north line makes with the line segment 𝐴𝐵.

Finally, it’s worth noting that we can use bearings to describe specific compass points. We start at north and travel in a clockwise direction. So, east is a bearing of 090. South would be a bearing of 180 degrees. West sits on a bearing of 270. And the north line itself represents a full turn; it’s a bearing of 360. We’ll now look at how we can use these facts to measure very simple bearings.

Find the bearing of 𝐵 from 𝐴.

Let’s begin by recalling what we understand by the word “bearing.” A bearing is a fancy way of measuring an angle. When we measure bearings, we remember three things. We first measure from north or from a north line. Once we’ve identified that north line, we measure in a clockwise direction. In navigation, we also use three digits to describe our bearing. Although depending on where you are in the world, when we’re not working with navigation problems, we’ll sometimes use two digits to represent a two-digit number or even include compass points.

The question wants us to find the bearing of 𝐵 from 𝐴, so we’re going to measure the bearing at 𝐴 using the rules we’ve given. Firstly, we identify the north line at 𝐴. We then travel in a clockwise direction until we hit the line segment that joins 𝐴 to 𝐵. According to our diagram, the angle that these two lines make with one another is 75 degrees. And so, if we were going to be following rule three, we would represent the bearing of 𝐵 from 𝐴 as 075. Now, of course, since this isn’t a navigation problem, the use of three digits isn’t quite as necessary. So, this might also be written as 75 degrees. The bearing of 𝐵 from 𝐴 then is 075 or 75 degrees.

In our next example, we’ll look at how to find the location of an object, given information about bearings from two different points.

The diagram shows the position of two lighthouses, A and B. A ship is on a bearing of 068 degrees from lighthouse A and a bearing of 295 degrees from lighthouse B. Mark the position of the ship on the diagram.

We’re given the location of this ship relative to two points, A and B. In fact, we’re told the location in terms of its bearings. So, let’s recall what we mean by a bearing. A bearing is a fancy way of measuring an angle, and we remember that we begin by measuring from the north line. We always measure in a clockwise direction from this north line round to the line segment we’re interested in. And, when necessary, when working with navigation, we try to use three digits or three-figure bearings. And so, a bearing of 068 is an angle of 68 degrees from the north line. So, we’ll begin by measuring this bearing, a bearing of 68 or 068 degrees from lighthouse A.

We first locate the north line at A. Because we’re looking to measure in a clockwise direction, we place the protractor as shown. By traveling up the north line, we see that the zero we’re looking for is in the outer row of numbers. And so, we travel in a clockwise direction around this outer row of numbers until we find an angle of 68 degrees. 68 degrees on our outer row of numbers is here. So next, we’re going to remove the protractor and join A to this point. We draw a single line segment from A through this point, as shown.

We’re now going to repeat this process for our second bearing; that’s a bearing of 295 degrees from lighthouse B. Once again, we begin by locating the north line. We’re now going to go in a clockwise direction from this north line. Note that we’re looking to measure a bearing of 295 degrees. And so, we have a little bit of a problem. If we were to measure in a clockwise direction and place our protractor as shown, we see that, in this direction, we can only go as far as 180 degrees. And so, we mark 180 degrees on our diagram and we rotate our protractor. Our job now is to figure out exactly how much further around we need to go to hit a bearing of 295 degrees.

Now what we know is that we’ve already traveled 180 degrees in a clockwise direction. So, let’s subtract 180 from 295. 295 minus 180 is 115. And so, we’re going to travel around. Once again, we’re using the outer row of numbers because we’re starting at zero, and we’re going to go in a clockwise direction round to 115. 115 degrees is here. Let’s remove the protractor and join this up with point B. Let’s draw a nice long line segment from B and through this point, as shown. We know that the ship sits on a bearing of 068 degrees from lighthouse A and 295 degrees from lighthouse B. This must mean the ship is located where our two lines representing these bearings overlap. That’s here. And so, we’ve marked with a cross the location of the ship on our diagram.

Note that there was another way we could’ve calculated the exact location of our second bearing. We know the angles around a point sum to 360 degrees. So, we could alternatively have subtracted 295 from 360 and then measured 065 or 65 degrees in a counterclockwise direction.

We’ll now look at something called back bearings or reverse bearings.

Find the bearing of 𝐴 from 𝐵.

We first recall that a bearing is simply a way of measuring an angle. We remember three things. We measure from the north line, and we always do so in a clockwise direction until we reach our line segment. In navigation, we also use three digits. So, for example, 73 degrees as a bearing would be 073. Now, the question here wants us to find the bearing of 𝐴 from 𝐵. This means we’re going to be measuring the bearing at 𝐵. So, let’s add a north line in. Now, since this diagram may not necessarily be to scale, we’re going to use some rules for working with parallel lines to deduce the bearing of 𝐴 from 𝐵.

Remember, we measure in a clockwise direction from our north line round to the line segment that joins 𝐴 to 𝐵. So, that’s this angle shown. And so, to calculate this angle, we’re going to carry the north line a little bit down from 𝐵. We know that our north lines must be parallel and that alternate angles are equal, so we can mark this angle on our diagram as being equal to the angle given; it’s 115 degrees. We also know that angles on a straight line sum to 180 degrees. So, this angle that I’ve marked along our north line is 180. It follows then that the bearing of 𝐴 from 𝐵 must be the sum of these two angles. It must be 180 plus 115. 180 plus 115 is 295. And so, we see the bearing of 𝐴 from 𝐵 is 295 degrees.

This is sometimes called a back bearing or a reverse bearing, and we can generalize this a little. We say that the difference between a bearing and its reverse bearing will always be 180 degrees. And so, given a bearing of 𝐴 from 𝐵, we find the bearing of 𝐵 from 𝐴 by either adding or subtracting 180 degrees. Whether we add or subtract will, of course, depend on the size of the original bearing. In this case, for example, we wouldn’t subtract 180 since 115 minus 180 is negative, and we know that we don’t work with negative bearings.

In our next example, we’ll look at how we can combine bearings with points on the compass.

A plane flies on the bearing shown. The control tower told the pilot to fly due west toward the airport. Which of the following is the angle the pilot should turn through? Is it (A) 137 degrees clockwise, (B) 90 degrees counterclockwise? Is it (C) 270 degrees counterclockwise, (D) 133 degrees counterclockwise, or (E) 223 degrees clockwise?

We’re given a path of a plane in our diagram and we’re told that, at some point, the control tower tells the pilot to fly due west. We know that relative to the north line, to travel west, we’d move to the left on our diagram. And so, we see that the plane is going to travel left on our diagram. Now, given the direction the plane is traveling, there are two ways the pilot could achieve this. Firstly, they could travel in a counterclockwise direction as shown. A slightly longer route will be to travel in a clockwise direction. Let’s begin by looking at the counterclockwise direction since this is slightly shorter.

Now, there are a number of ways we can find this angle. One way is to add a north line at the location of the pilot. We then know the two north lines in our diagram are parallel. So, we can add 43 degrees here, since we know that corresponding angles are equal. We also know that the north line and the west line are perpendicular. They meet at an angle of 90 degrees. And so, we can calculate the angle that the pilot turns through by adding 90 and 43. 90 plus 43 is 133 degrees. And so, the pilot could turn 133 degrees counterclockwise. And that’s option (D).

But remember, we said that the pilot could have turned in the opposite direction. They could have traveled in a clockwise direction. So, how could we have calculated this angle? Well, by extending the west line and recalling that the north line and west line are perpendicular, we can draw a right-angle triangle, as shown. We know that angles in a triangle sum to 180 degrees. So, we can find the third angle in this right-angle triangle by subtracting 90 and 43 from 180 to get 47 degrees.

We know that angles on a straight line sum to 180 degrees. And so, had the pilot turned on a clockwise direction, the angle would’ve been calculated by adding 180 to 47 to get 227 degrees. And so, this isn’t one of our options. But we could have said that the pilot needed to turn on an angle of 227 degrees clockwise.

In our final example, we’ll look at how we can use a little bit of geometry to solve bearings problems.

Two roads 𝑂𝐴 and 𝑂𝐵 are shown in the diagram. The roads meet at an angle of 112 degrees. Jackie is at point 𝑂, the intersection of the two roads. She walks towards 𝐴. On what bearing does she walk?

She is currently at point 𝑂, and she wants to walk towards point 𝐴. And so, we’re going to need to calculate the bearing of 𝐴 from 𝑂. We recall that when we measure and draw bearings, we begin by looking at the north line. We then measure in a clockwise direction until we hit the line segment we’re interested in. So that’s the line segment 𝑂𝐴. So, how are we going to calculate this angle? We know that angles on a straight line add to 180 degrees. And so, we can find the measure of the angle we’re looking for by subtracting the given angle, 112 degrees, from 180. 180 minus 112 is 68. The angle that we’re looking for then is 68 degrees.

And remember, when we’re working, particularly, with navigation problems, we tend to use three-figure bearings. So, as a bearing, 68 degrees is 068. Jackie must walk on a bearing of 068 degrees towards 𝐴.

In this video, we saw that a bearing is a fancy way of measuring an angle, and it’s often used in navigation. We saw that to measure and draw bearings, we begin by identifying the north line. Once we’ve identified that north line, we measure our angle in a clockwise direction round to the line segment we’re interested in. And that we tend to use three digits to represent our bearings. We have problems that don’t involve any navigation. Every so often, this might just be simply represented using a two-digit number or, alternatively, by referencing the compass points.

Finally, we saw that the difference between a bearing and its reverse bearing or back bearing is always 180 degrees. So, given a bearing of 𝐴 from 𝐵, we calculate the bearing of 𝐵 from 𝐴 by either adding or subtracting 180 degrees and making sure that we don’t get a negative value.