## What you need to know

Bearings are a way of expressing the angle between two objects, and there is a specific set of rules about how bearings should be calculated and expressed. The reason for having these rules is probably because sailors needed a way to communicate their locations to each other, but that can get confusing when you’re in the middle of the ocean and every direction looks the same. One way to fix this issue was to say that all angles should be measured from the North line going clockwise.

For example, suppose we have two points, A and B, and we are asked for the bearing of B from A.

**Note:** the terminology “B from A” is always used, as opposed to “A to B”. This is one of those rules that we mentioned, and it just ensures everyone is speaking the same language. What it means for our calculations is that we want to find direction of B if we were looking from A, so A is where we choose to measure our angle from.

So, we have our two points and a North line coming off both. The bearing of B from A is measured from the North line going clockwise until we hit the straight line which joins points A and B. The angle is measured to be 110 degrees, and that is the bearing of B from A.

Another rule is that whenever we are expressing a bearing, it should be three digits. This is natural for 110 degrees, but if the resulting angle was 6 degrees or 52 degrees, then the bearings should be expressed like 006\degree and 052\degree respectively. For this reason, you will often hear them referred to as **three-figure bearings. **

In general, you will either be asked to use an existing diagram to determine the bearing between two points (using a protractor or facts given in the question) or, given the size of a bearing and the distance between two points, construct a picture like the one above.

In summary, the rules are:

1. Always measure bearings from the North line going clockwise;

2. Always express your answers as three-figure bearings;

3. The bearing of B from A refers to the angle which is measured from the point A looking at B.

**Example: **Two boats A and B are 5km apart, and the bearing of B from A is 256\degree. Using the scale 1cm:1km, construct a diagram showing the relative positions of points A and B.

As we’re finding the bearing of B from A, we’re going to measuring our angle at the point A, so we’ll start by drawing the point A with a North line and measuring an angle of 256\degree going clockwise from it.

Then, as A and B are 5km apart, we will need to make the line from A to B (going along the bearing we’ve determined) 5cm long.

The result of this (and the completed diagram) is below.

**Example:** The diagram below shows the bearing of B from A. Find the bearing of A from B.

Now, we can’t measure the angle because the diagram is not drawn accurately. So, we must think cleverly about this one. We will use the fact that both North lines are parallel. If we extend the line AB a little further past the point B, then the angle formed by the North line at B and this new extension to the line AB and the bearing of B from A are **corresponding** angles (also known as an “F angle”). So, from our knowledge or parallel lines, we know that they must be equal.

Now we are almost there, but not quite. When measuring the bearing of A from B, we need to measure the clockwise angle from the North line all the way around to the line **between** A and B (i.e., we need to go further than the 94 degrees).

Fortunately, the rest is easy. The remaining portion of the angle is just a straight line, so the bearing of A from B is

94 + 180 = 274\degree

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### Example Questions

1) A boat and a lighthouse are 70 miles apart. The bearing of the lighthouse from the boat is 051\degree. Using the scale 1cm:10 miles, construct a diagram showing the relative positions of the lighthouse and the boat.

Let the lighthouse be L and the boat be B. As we’re finding the bearing of L from B, we’re going to measuring our angle at the point B, so we’ll start by drawing the point B with a North line and measuring an angle of 051\degree going clockwise from it.

Then, as B and L are 70 miles apart, we will need to make the line from A to B (going along the bearing we’ve determined) 7cm long. The final diagram is below.

2) The diagram below shows the bearing of A from B. Find the bearing of B from A.

We can’t simply measure the angle, since the picture is not drawn accurately. Instead, we will use the fact that the two North lines are parallel to one another.

Firstly, recognise that we can find the other angle around the point B by subtracting 295 from 360.

360 - 295 = 65 \degree

Then, because the two North lines are parallel, we can say that the bearing of B from A and the 65\degree angle we just found are **interior **(sometimes co-interior, or allied, depending on what your teacher likes). From our facts about angles in parallel lines, we know the two angles (marked with red below) must add to 180.

So, we get:

\text{Bearing of B from A } = 180 - 65 = 115\degree.

### Worksheets and Exam Questions

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#### Bearings Q1

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## Bearings Worksheets

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