GCSE · Maths · AQA · Spec 8300

Measure line segments and angles; bearings

"The café is 2 km away" doesn't tell you which way to walk. A bearing gives direction as one exact number; a scale turns map centimetres into real kilometres.

Bearings · Start here

Where is B from A?

Angle between North and AB 60°. Length AB 5.0 cm. Classification: Measured at A · from North · clockwise. Relationship: Bearing of B from A: if B is east of the North line, it is the angle shown. If B is west of it, it is 360° minus the angle shown.

Angle between North and AB60°Length AB5.0 cmdrag B round A

Measured at A · from North · clockwise

Bearing of B from A: if B is east of the North line, it is the angle shown. If B is west of it, it is 360° minus the angle shown.

Drag B slowly all the way round A, starting from North. East of the North line, the angle shown IS the bearing. Keep going past South and watch the angle start shrinking again, even though you are still turning clockwise. Park B so the angle reads 60° on the right, then 60° on the left: same reading, but bearings of 060° and 300°. The side of the North line decides. Notice too that AB stays 5.0 cm wherever B goes: a bearing gives the direction, not the distance.

Exam line: Write every bearing with three figures: 60° clockwise from North is 060°, and 5° is 005°.

Watch out: A protractor only measures up to 180°. For a point west of the North line, measure the angle from North going the other way, then take it away from 360°.

Compass points as bearings

One clockwise turn, eight landmarks

North is 000° (and a full turn brings you back to it at 360°). South, halfway round, is 180°. Drag the other six compass points to their bearings, then check.

NE — North-eastE — EastSE — South-eastSW — South-westW — WestNW — North-west

Measuring the angle

Two scales, one right answer

You want the bearing of C from A. You put the centre of your protractor exactly on A and line up its 0 with the North line at A. The line AC crosses the edge where one scale says 70 and the other says 110. C is somewhere between North and East of A.

Which is closest to what you would write?

110°, because you read the outside scale.070°, because I count up from the 0 that sits on the North line.Either one works, because both numbers are on the protractor at that spot.70°, because you always read the inside scale.

How sure are you?

I'm sureI think soI'm guessing

110°, because you read the outside scale.

This is a common misconception

Neither scale is automatically the right one. Turn the protractor round and the two scales swap jobs. Here the outside scale is counting from the South end of the line, not from North. And a quick estimate catches it: C is between North (000°) and East (090°), so the answer can't be 110°.

070°, because I count up from the 0 that sits on the North line.

This matches the evidence

Yes. The scale whose 0 is on your starting line is the one that measures from that line, and a bearing starts at North. It also passes the estimate test: between North and East means between 000° and 090°, and 070° is. Written with three figures, it is 070°.

Either one works, because both numbers are on the protractor at that spot.

This is a common misconception

They are two different angles. The line AC makes 70° with North and 110° with South, and together they make the 180° of the straight North–South line. A bearing is measured from North, so only one of them is the bearing.

70°, because you always read the inside scale.

This is a common misconception

You got 70 this time, but the rule is wrong, so it will let you down. Put the protractor down the other way round and the inside scale starts at South instead. The rule isn't inside or outside: it's start from the 0 on your North line. Then write it with three figures: 070°.

Find the 0 that sits on the North line and count up from it. Then sanity-check your reading against the compass points the line lies between.

Exam line: Before you read the protractor, estimate: which two compass points is the line between? Your reading must land in that range.

Maps and scale drawings

1 cm on the drawing is 2 km for real

the length on the drawing: 6/16. Length on the drawing 3.0 cm. Real distance 6 km

Length on the drawing3.0 cmReal distance6 km

The bar is a line on a scale drawing where 1 cm represents 2 km. Drag it and watch the two lengths move together: every centimetre you add brings 2 more kilometres with it. Try it both ways. How long is a 6.5 cm line in real life? And if two places are 11 km apart, how long is the line on the drawing?

Watch out: Drawing to real: multiply by 2. Real to drawing: divide by 2. If your drawing length comes out bigger than the real distance in km, you went the wrong way.

Check the answer

Find the line that goes wrong

A map has a scale of 1 cm to 5 km. Lighthouse L is between North and West of harbour H. Find the distance and the bearing of L from H.

A student's answer — which line goes wrong?

Measure HL on the map with a ruler, starting from 0: it is 3.8 cm.
Real distance = 3.8 × 5 = 19 km.
Put the centre of the protractor on H, with its 0 on the North line at H.
The angle between North and the line HL is 48°.
So the bearing of L from H is 048°.
Answer: L is 19 km from H on a bearing of 048°.

Exam line: Before you write a bearing down, ask: is the point east or west of the North line? West means the bearing is between 180° and 360°.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Give any direction as one exact number, and turn centimetres on a map into real distances.

What you need to know

A bearing is an angle measured at the starting point, from North, turning clockwise.
Bearings are always written with three figures: 7° is 007°, 60° is 060°.
The compass points as bearings: N 000°, NE 045°, E 090°, SE 135°, S 180°, SW 225°, W 270°, NW 315°.
"The bearing of B from A" means stand at A and measure there.
For a point west of the North line: bearing = 360° − the angle between North and the line.
On a map or scale drawing: real distance = drawing length × scale; drawing length = real distance ÷ scale.

The big picture

A bearing is an angle with fixed rules: measure it at the point you are going FROM, start at North, turn clockwise, and write three figures, so 60° is 060°. N, E, S and W sit at 000°, 090°, 180° and 270°, with the in-between points 45° further round each time. A protractor stops at 180°, so for a point west of the North line, measure from North the other way and subtract from 360°. On a map or scale drawing, measure with a ruler, then multiply by the scale for the real distance, or divide to go back.

Key points

1Start point, North, clockwise, three figures: those four rules define every bearing.

2Each of the eight compass points is 45° further round than the one before.

3Read the protractor scale whose 0 is on your starting line, and check the reading against an estimate.

4A semicircle protractor stops at 180°, so bearings over 180° are found as 360° minus the anticlockwise angle from North.

5A bearing gives direction only; to fix a position you also need a distance.

6Measure lengths with a ruler from 0, then use the scale to convert between drawing and real life.

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Worked example

Problem

On a scale drawing, 1 cm represents 4 km. Port P is marked. Ship Q is 14 km from P on a bearing of 240°. Explain how to mark Q accurately.

⚠ Watch out

Measuring the angle correctly and then writing it straight down as the bearing when the point is west of the North line. A point on the west side always has a bearing between 180° and 360°, so take the anticlockwise angle from North away from 360°.

🧠

Memory hook

Stand at the start, face North, turn like a clock hand, say it in three.

✓

Check yourself

No protractor needed: what is the bearing of north-west? And if B is on a bearing of 290° from A, is B east or west of A, and which compass point is it nearest?

Flashcards

(9)

What is a bearing?

An angle measured at the starting point, from North, turning clockwise, written with three figures.

Bearings of N, E, S and W

000°, 090°, 180°, 270°.

Bearings of NE, SE, SW and NW

045°, 135°, 225°, 315°.

"The bearing of X from Y": where do you measure?

At Y, the point after the word "from". Draw the North line there.

How do you write an angle of 8° as a bearing?

008°. Bearings always use three figures, so small angles get zeros in front.

The point is west of the North line. How do you turn your protractor reading into a bearing?

Measure the angle from North going anticlockwise, then work out 360° minus that angle.

A protractor has two scales. Which do you read?

The one whose 0 sits on your starting line. For a bearing, that's the North line.

Does a bearing tell you how far away something is?

No. It only gives direction. You need a distance as well to fix a position.

Scale "1 cm represents k km": how do you convert?

Drawing to real: multiply the cm by k. Real to drawing: divide the km by k.

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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One clockwise turn, eight landmarks

Two scales, one right answer

Find the line that goes wrong

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