GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher

Measuring lines, angles and bearings

“Meet me at the café, it's 3 km away.” Great, but which way? Distance alone leaves you lost, and so does direction alone.

Maths · Bearings

Where is B from A?

Angle from the North line to AB 35°. Classification: Measured at A · from North · clockwise. Relationship: Bearing of B from A: if B is to the right of the North line, it is the angle shown, written with three figures (35° is 035°). If B is to the left, the angle shown is the anticlockwise one, so the bearing is 360° minus it: 360° − 114° = 246°. Same answer another way: 246° is 180° plus a further 66°.

Angle from the North line to AB35°drag B round A

Measured at A · from North · clockwise

Bearing of B from A: if B is to the right of the North line, it is the angle shown, written with three figures (35° is 035°). If B is to the left, the angle shown is the anticlockwise one, so the bearing is 360° minus it: 360° − 114° = 246°. Same answer another way: 246° is 180° plus a further 66°.

Drag B round A. Stand at A, face the North line, then turn clockwise until you face B. That turn is the bearing.

Exam line: Clockwise from North, measured at the starting point, written with three figures: 35° becomes 035°.

Watch out: The readout never shows more than 180°. When B is to the left of the North line it shows the anticlockwise angle, so take it away from 360° to get the bearing.

Which point?

Whose bearing is it anyway?

A diver is underwater. A boat sits on the surface. Someone says: “The bearing of the boat from the diver is 040°.”

Where was that 040° angle measured, and which way did it turn? Pick the idea closest to yours.

At the boat. It's the boat's bearing, so I measure the angle at the boat.At the diver, but I turn whichever way round is shorter to reach the boat.At the diver. Draw the North line at the diver first, then turn clockwise until I face the boat.It doesn't matter. The angle between the two is the same whichever end I measure from.

How sure are you?

I'm sureI think soI'm guessing

At the boat. It's the boat's bearing, so I measure the angle at the boat.

This is a common misconception

Easy to do, because the boat is the thing being described. But the point after “from” is where you stand. The bearing of the boat from the diver is measured at the diver. Measuring at the boat would give the bearing of the diver from the boat, which is a different angle.

At the diver, but I turn whichever way round is shorter to reach the boat.

This is a common misconception

Right place, wrong rule. A bearing is always the clockwise turn from North, even when the anticlockwise turn is shorter. If the clockwise angle is more than 180°, you can measure the anticlockwise angle with your protractor and take it away from 360°. The answer you write is still the clockwise one.

At the diver. Draw the North line at the diver first, then turn clockwise until I face the boat.

This matches the evidence

That's the whole method. The diver is the point after “from”, so that is where North gets drawn and where the angle sits. Then you turn clockwise to the boat and write the angle with three figures.

It doesn't matter. The angle between the two is the same whichever end I measure from.

This is a common misconception

This is the trap. The bearing of the boat from the diver and the bearing of the diver from the boat are different angles, because each one uses the North line drawn at its own starting point.

The bearing of X from Y is measured at Y: stand at Y, face North, turn clockwise to X.

Predict, then check

Commit to an answer first. The reveal is worth the wait.

A lighthouse keeper sees a boat on a bearing of 112° from the lighthouse. Is that enough to say exactly where the boat is?

Worked example · calculating a bearing

Problem

B is on a bearing of 065° from A. On this sketch (not to scale), work out the bearing of A from B, giving a reason for each step.

Scales

Fill in the missing steps

Turn the scale 1 cm = 25 km into a ratio, then use a 1 : 300 000 map to find a real distance.

The scale is 1 cm = 25 km. A ratio needs the same unit on both sides, so change the kilometres into centimetres.
missing step

Which line is step 2?

25 km = 25 000 cm (25 × 1000)25 km = 2 500 000 cm (25 × 100 000)25 km stays as 25, so the ratio is 1 : 25

Scale drawing

Which line ruins the drawing?

Sam is planning an accurate scale drawing of a straight path that is 6 km long. One line of the plan is wrong. Find it.

Sam's plan — which line goes wrong?

Choose a scale that fits the page and lets me measure to the nearest millimetre: 1 : 200 000.
Put the real length in the same units as the scale: 6 km = 600 000 cm.
Multiply by the scale factor to get the length to draw: 600 000 × 200 000.
So I draw the path 120 000 000 000 cm long.
Use a ruler for the length and a protractor for any angle.
Write the scale, 1 : 200 000, next to the diagram.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Which way, how far, and measured from where?

What you need to know

A bearing is an angle measured from North in the clockwise direction.
Write it with three figures, so 35° becomes the bearing 035°.
Have a goDev measures 7° clockwise from North and announces, “The bearing is 7°.” His teacher sighs. Fix Dev's answer.
007°
A bearing always has three figures, so a one-digit angle needs two zeros in front of it.
The bearing of B from A is measured at A, from the North line drawn at A.
That makes the bearing of B from A a different angle from the bearing of A from B.
Have a goMo says: “The bearing of the library from the school is 125°, so the bearing of the school from the library must be 125° too.” Is Mo right?
No. It is a different angle.
The first bearing is measured at the school and the second at the library, each from its own North line, so swapping the points changes the angle.
On an accurate diagram, use a protractor; for a clockwise angle over 180°, measure the anticlockwise angle and subtract it from 360°.
Have a goA protractor shows 70° anticlockwise from North. What is the bearing?
290°
The clockwise angle is over 180°, so subtract the anticlockwise one from 360°: 360° − 70° = 290°. Reading it as 070° would turn the wrong way.
A bearing alone gives only a line; a distance, or a second bearing from another point, fixes the position.
North lines at different points are parallel, so angle facts let you calculate a bearing on an unscaled diagram.
A scale like 1 cm = 25 km becomes the ratio 1 : 2 500 000 once both sides use the same units.
To draw, divide real lengths by the scale factor; to read a map, multiply the map length by it.
Draw lengths with a ruler and angles with a protractor, and write the scale next to the diagram.

The big picture

A bearing is a clockwise turn from North, measured at the point you start from and written with three figures. A single bearing gives only a line, so a distance or a second bearing fixes a position. Scale drawings turn real lengths into drawn ones by dividing by the scale factor.

Key points

1A bearing is a clockwise angle from North, measured at the starting point and written with three figures.

2The bearing of B from A and the bearing of A from B are different angles.

3When the clockwise angle is over 180°, find it as 360° minus the anticlockwise angle.

4One bearing gives a line; a distance or a second bearing from another point fixes the position.

5On a scale drawing, divide real lengths by the scale factor, draw accurately and write the scale beside the diagram.

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Worked example

Problem

Draw a line from a point P on a bearing of 255°, using a protractor that only goes up to 180°.

⚠ Watch out

Reading the protractor without checking which side of the North line the point is on. To the left of it, the protractor shows the anticlockwise angle, so the bearing is 360° minus that reading.

🧠

Memory hook

Stand, face North, turn clockwise. And “from” tells you where to stand.

✓

Check yourself

Cover the page: where do you stand to find “the bearing of the school from the shop”, which way do you turn, and why isn't one bearing enough to place the school?

Flashcards

(15)

What is a bearing?

An angle measured in degrees from North in the clockwise direction, written with three figures.

How do you write an angle of 35° as a bearing?

035°. A bearing always has three figures.

Where is the bearing of B from A measured?

At A. Draw the North line at A, then turn clockwise to B.

Is the bearing of A from B the same as the bearing of B from A?

No. They are different angles, each measured at its own starting point.

What does the word “measure” signal on an accurate diagram?

Use an instrument: measure the bearing with a protractor.

The clockwise angle is over 180°. How can you find the bearing?

Measure the anticlockwise angle from North and subtract it from 360°.

Why does one bearing not fix a position?

It gives only a line, because every point along it has the same bearing.

What fixes the position of a point that has a bearing?

A distance along the line (using a scale), or a second bearing from another known point.

Why are the North lines at two different points parallel?

North is always the same direction.

Which angle facts help calculate a bearing on an unscaled diagram?

Corresponding angles equal, alternate angles equal, angles on a straight line 180°, angles at a point 360°, angles in a triangle 180°. Give a reason for each step.

What must you do before writing a scale such as 1 cm = 25 km as a ratio?

Convert both sides to the same units. Then 1 cm = 25 km becomes 1 : 2 500 000.

How can you check a conversion between a map and real life?

Multiply in one direction and divide in the other. You should arrive back at the start.

To get the length to draw on a scale diagram, do you multiply or divide the real length by the scale factor?

Divide.

What else must a scaled drawing show besides accurate lengths and angles?

The scale, written next to the diagram. Use a ruler for lengths and a protractor for angles.

A scale is written with no units. What units does it use?

The same units for both values.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Whose bearing is it anyway?

Fill in the missing steps

Which line ruins the drawing?

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