GCSE Maths: Bearings: measuring, drawing and back bearings (Foundation)
Part of the GCSE Maths loci, bearings and vectors chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Bearings: measuring, drawing and back bearings
A bearing is an angle that tells you the direction of one place from another. Pilots, sailors and walkers use bearings because a direction such as “north-east” is not exact enough. Every bearing follows three rules.
- Measure from north. The angle always starts at the north line (straight up the page).
- Measure clockwise. Turn the way the hands of a clock turn, so east is 90°.
- Give three figures. A bearing of 5° is written 005° and 45° is written 045°. Bearings go from 000° to 360°.
Reading a bearing. To find the bearing of B from A, stand at A and look at B. Draw a north line at A. Then turn clockwise from the north line until you point to B. The angle you turn through is the bearing. Be careful with the word “from”. The bearing of B from A starts at A. It is a different bearing from the bearing of A from B.
Drawing a bearing. To draw a line from P on a bearing of 135°: (1) draw a north line through P; (2) put the protractor on P with its zero on the north line; (3) turn clockwise to 135° and mark a dot; (4) draw a line from P through the dot. A bearing between 180° and 360° goes past south. You can measure 360° minus the bearing anticlockwise from north instead.
North lines are parallel. The north line at every point points the same way, so the north lines at two different points are parallel. This is the key to all bearings angle problems. Always draw a north line at each point you measure from.
Back bearings. If the bearing of B from A is 070°, what is the bearing of A from B (the way back)? Draw north lines at A and B. They are parallel lines and the line AB crosses both. The angle at B between the north line and the line back to A is a co-interior angle with 70° (they add to 180°), so it is 180° − 70° = 110°. But we need the angle measured clockwise, so take it from a full turn: 360° − 110° = 250°.
The shortcut. The way back is the opposite direction, which is half a turn (180°) away. So back bearing = bearing ± 180°. If the bearing is less than 180°, add 180°. If it is 180° or more, subtract 180°. This keeps the answer between 000° and 360°.
Key ideas
- A bearing is measured from north, clockwise, with three figures.
- The bearing of B from A: stand at A, face north, turn clockwise to B.
- Draw a north line at every point you measure from. North lines are parallel.
- Back bearing: add 180° if the bearing is less than 180°, subtract 180° if it is 180° or more.
- Co-interior angles between parallel north lines add up to 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Start at the north line, then turn clockwise until you reach the line. Read the number on the dial.
- Find north. Bearings always start from north, which is straight up the page.North = 000°
- Turn clockwise to the line. Follow the dial round the way a clock hand moves. The tick marks are 5° apart and the numbers are 30° apart.The line crosses the dial at 70°
- Write three figures. A bearing is always written with three figures, so put zeros in front of a small number.Bearing = 070°
Check. A line at 70° is less than 180°, so it points to the east side. This agrees with the picture. ✓
Watch out. Do not measure anticlockwise. A line pointing to the west side has a bearing bigger than 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at P, measure the bearing clockwise from north, then draw the line to the right length.
- Mark P and draw a north line through it. Every bearing is measured from north. Use a ruler to draw a line pointing straight up the page.Point P with a north line
- Place the protractor. Put the centre of the protractor on P and its 0° line along the north line.Use the protractor the usual way round: start at 000° (north) and turn clockwise to 135°.
- Make a mark and draw the line. Mark a small dot at 135°. Draw a line from P through the dot.Line at 135° from north
- Measure 5 cm along the line and label Q. Use a ruler. Label the end of the line.
Check. The line should point to the east half of the page, because 135° is less than 180°. ✓
Watch out. Always start from the north line and measure clockwise. Do not measure from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of A from B is the back bearing of the bearing of B from A: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines both point the same way, so they are parallel. The line AB crosses both of them.Bearing of B from A = 070°
- Find the angle at B between the north line and BA. Angles between parallel lines and on the same side of the crossing line are co-interior and add up to 180°.180° − 070° = 110°
- Turn it into a bearing. A bearing is measured clockwise from north, but this angle is measured the other way round. Take it away from a full turn of 360°.360° − 110° = 250°
- Write it with three figures and check with the shortcut. Going back is the exact opposite way, which is half a turn (180°) different from the forward bearing.70° + 180° = 250°. Back bearing = 250°
Check. Reverse it again: 250° − 180° = 70°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of A from B is the back bearing of the bearing of B from A: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines are parallel. The line AB crosses both of them.Bearing of B from A = 290°
- Find the angle between the north-south line at A and AB. The bearing is more than 180°, so the line has gone past south. The extra angle past south is the bearing minus 180°.290° − 180° = 110°
- Use alternate angles at B. Alternate angles (a Z shape) between parallel lines are equal. So the angle at B between the north line and BA is the same size, and it is measured clockwise from north.Angle = 110°
- Write it with three figures and check with the shortcut. Going back is half a turn different. Because the forward bearing is more than 180°, subtract 180°.290° − 180° = 110°. Back bearing = 110°
Check. Reverse it again: 110° + 180° = 290°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find the bearing of A from B (the back bearing). Then angle ABC is the gap between the bearings of A and C, both measured from the north line at B.
- Find the back bearing. The bearing of A from B is half a turn from the bearing of B from A: 40° is less than 180°, so add 180°.40° + 180° = 220°
- Mark the bearings at B. At B the north line is parallel to the one at A. The line BA is at 220° and the line BC is at 110°, both measured clockwise from north.BA = 220°, BC = 110°
- Find the angle between them. Angle ABC is the difference between the two bearings.220° − 110° = 110°
- Write the answer. This is the angle inside the triangle.Angle ABC = 110°
Check. Another way: the ship turns right by 110° − 40° = 70°. The angle inside the corner is 180° − 70° = 110°. ✓
Watch out. Draw the north line at B (the point you are working from), not just at A.
Watch out
Do not forget the north line at the second point. A bearing of B from A uses the north line at A. A bearing of A from B uses a different north line, at B. Also remember to write three figures (so 45° is 045°), and measure clockwise, even if the line is on the left of the page.Exercise 22G Bearings: reading, drawing and back bearings
Non-calculatorKey words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Count clockwise from north in steps of 90° (a quarter turn) or 45° (an eighth of a turn).
- Start at north. North is 000°.N = 000°
- Turn clockwise. North is the starting line, so there is no turn.0 × 45° = 0°
- Write three figures. 000°
Check. The compass rose in this section shows the same bearing. ✓
Watch out. East is 090° (not 180°) because you turn a quarter of a full turn.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Count clockwise from north in steps of 90° (a quarter turn) or 45° (an eighth of a turn).
- Start at north. North is 000°.N = 000°
- Turn clockwise. East is 2 eighth-turns of 45° clockwise from north.2 × 45° = 90°
- Write three figures. 090°
Check. The compass rose in this section shows the same bearing. ✓
Watch out. East is 090° (not 180°) because you turn a quarter of a full turn.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Count clockwise from north in steps of 90° (a quarter turn) or 45° (an eighth of a turn).
- Start at north. North is 000°.N = 000°
- Turn clockwise. South is 4 eighth-turns of 45° clockwise from north.4 × 45° = 180°
- Write three figures. 180°
Check. The compass rose in this section shows the same bearing. ✓
Watch out. East is 090° (not 180°) because you turn a quarter of a full turn.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Count clockwise from north in steps of 90° (a quarter turn) or 45° (an eighth of a turn).
- Start at north. North is 000°.N = 000°
- Turn clockwise. West is 6 eighth-turns of 45° clockwise from north.6 × 45° = 270°
- Write three figures. 270°
Check. The compass rose in this section shows the same bearing. ✓
Watch out. East is 090° (not 180°) because you turn a quarter of a full turn.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Count clockwise from north in steps of 90° (a quarter turn) or 45° (an eighth of a turn).
- Start at north. North is 000°.N = 000°
- Turn clockwise. North-east is 1 eighth-turn of 45° clockwise from north.1 × 45° = 45°
- Write three figures. 045°
Check. The compass rose in this section shows the same bearing. ✓
Watch out. East is 090° (not 180°) because you turn a quarter of a full turn.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Count clockwise from north in steps of 90° (a quarter turn) or 45° (an eighth of a turn).
- Start at north. North is 000°.N = 000°
- Turn clockwise. South-west is 5 eighth-turns of 45° clockwise from north.5 × 45° = 225°
- Write three figures. 225°
Check. The compass rose in this section shows the same bearing. ✓
Watch out. East is 090° (not 180°) because you turn a quarter of a full turn.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A bearing needs three figures. Put zeros in front of a number with fewer than three digits.
- Count the digits. 8 has 1 digit.8
- Add zeros at the front. Make the number up to three figures.008°
Check. Bearings always have three figures, so 000, 045, 120, 359 are all fine. ✓
Watch out. 045° is not the same as 45° written without a zero: it is the same angle, but the three-figure form is correct for bearings.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A bearing needs three figures. Put zeros in front of a number with fewer than three digits.
- Count the digits. 72 has 2 digits.72
- Add zeros at the front. Make the number up to three figures.072°
Check. Bearings always have three figures, so 000, 045, 120, 359 are all fine. ✓
Watch out. 045° is not the same as 45° written without a zero: it is the same angle, but the three-figure form is correct for bearings.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A bearing needs three figures. Put zeros in front of a number with fewer than three digits.
- Count the digits. 5 has 1 digit.5
- Add zeros at the front. Make the number up to three figures.005°
Check. Bearings always have three figures, so 000, 045, 120, 359 are all fine. ✓
Watch out. 045° is not the same as 45° written without a zero: it is the same angle, but the three-figure form is correct for bearings.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A bearing needs three figures. Put zeros in front of a number with fewer than three digits.
- Count the digits. 120 has 3 digits.120
- Add zeros at the front. Make the number up to three figures.120°
Check. Bearings always have three figures, so 000, 045, 120, 359 are all fine. ✓
Watch out. 045° is not the same as 45° written without a zero: it is the same angle, but the three-figure form is correct for bearings.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Start at the north line, then turn clockwise until you reach the line. Read the number on the dial.
- Find north. Bearings always start from north, which is straight up the page.North = 000°
- Turn clockwise to the line. Follow the dial round the way a clock hand moves. The tick marks are 5° apart and the numbers are 30° apart.The line crosses the dial at 35°
- Write three figures. A bearing is always written with three figures, so put zeros in front of a small number.Bearing = 035°
Check. A line at 35° is less than 180°, so it points to the east side. This agrees with the picture. ✓
Watch out. Do not measure anticlockwise. A line pointing to the west side has a bearing bigger than 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Start at the north line, then turn clockwise until you reach the line. Read the number on the dial.
- Find north. Bearings always start from north, which is straight up the page.North = 000°
- Turn clockwise to the line. Follow the dial round the way a clock hand moves. The tick marks are 5° apart and the numbers are 30° apart.The line crosses the dial at 110°
- Write three figures. A bearing is always written with three figures, so put zeros in front of a small number.Bearing = 110°
Check. A line at 110° is less than 180°, so it points to the east side. This agrees with the picture. ✓
Watch out. Do not measure anticlockwise. A line pointing to the west side has a bearing bigger than 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Start at the north line, then turn clockwise until you reach the line. Read the number on the dial.
- Find north. Bearings always start from north, which is straight up the page.North = 000°
- Turn clockwise to the line. Follow the dial round the way a clock hand moves. The tick marks are 5° apart and the numbers are 30° apart.The line crosses the dial at 205°
- Write three figures. A bearing is always written with three figures, so put zeros in front of a small number.Bearing = 205°
Check. A line at 205° is more than 180°, so it points to the west side. This agrees with the picture. ✓
Watch out. Do not measure anticlockwise. A line pointing to the west side has a bearing bigger than 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Start at the north line, then turn clockwise until you reach the line. Read the number on the dial.
- Find north. Bearings always start from north, which is straight up the page.North = 000°
- Turn clockwise to the line. Follow the dial round the way a clock hand moves. The tick marks are 5° apart and the numbers are 30° apart.The line crosses the dial at 318°
- Write three figures. A bearing is always written with three figures, so put zeros in front of a small number.Bearing = 318°
Check. A line at 318° is more than 180°, so it points to the west side. This agrees with the picture. ✓
Watch out. Do not measure anticlockwise. A line pointing to the west side has a bearing bigger than 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at P, measure the bearing clockwise from north, then draw the line to the right length.
- Mark P and draw a north line through it. Every bearing is measured from north. Use a ruler to draw a line pointing straight up the page.Point P with a north line
- Place the protractor. Put the centre of the protractor on P and its 0° line along the north line.Use the protractor the usual way round: start at 000° (north) and turn clockwise to 60°.
- Make a mark and draw the line. Mark a small dot at 60°. Draw a line from P through the dot.Line at 60° from north
- Measure 6 cm along the line and label Q. Use a ruler. Label the end of the line.
Check. The line should point to the east half of the page, because 60° is less than 180°. ✓
Watch out. Always start from the north line and measure clockwise. Do not measure from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at P, measure the bearing clockwise from north, then draw the line to the right length.
- Mark P and draw a north line through it. Every bearing is measured from north. Use a ruler to draw a line pointing straight up the page.Point P with a north line
- Place the protractor. Put the centre of the protractor on P and its 0° line along the north line.Use the protractor the usual way round: start at 000° (north) and turn clockwise to 135°.
- Make a mark and draw the line. Mark a small dot at 135°. Draw a line from P through the dot.Line at 135° from north
- Measure 6 cm along the line and label Q. Use a ruler. Label the end of the line.
Check. The line should point to the east half of the page, because 135° is less than 180°. ✓
Watch out. Always start from the north line and measure clockwise. Do not measure from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at P, measure the bearing clockwise from north, then draw the line to the right length.
- Mark P and draw a north line through it. Every bearing is measured from north. Use a ruler to draw a line pointing straight up the page.Point P with a north line
- Place the protractor. Put the centre of the protractor on P and its 0° line along the north line.The bearing is more than 180°, so you go past south. Either measure 250° clockwise with a full-circle protractor, or measure 110° anticlockwise from north (because 360° − 250° = 110°).
- Make a mark and draw the line. Mark a small dot at 250°. Draw a line from P through the dot.Line at 250° from north
- Measure 6 cm along the line and label Q. Use a ruler. Label the end of the line.
Check. The line should point to the west half of the page, because 250° is more than 180°. ✓
Watch out. Always start from the north line and measure clockwise. Do not measure from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at P, measure the bearing clockwise from north, then draw the line to the right length.
- Mark P and draw a north line through it. Every bearing is measured from north. Use a ruler to draw a line pointing straight up the page.Point P with a north line
- Place the protractor. Put the centre of the protractor on P and its 0° line along the north line.The bearing is more than 180°, so you go past south. Either measure 300° clockwise with a full-circle protractor, or measure 60° anticlockwise from north (because 360° − 300° = 60°).
- Make a mark and draw the line. Mark a small dot at 300°. Draw a line from P through the dot.Line at 300° from north
- Measure 6 cm along the line and label Q. Use a ruler. Label the end of the line.
Check. The line should point to the west half of the page, because 300° is more than 180°. ✓
Watch out. Always start from the north line and measure clockwise. Do not measure from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of P from Q is the back bearing of the bearing of Q from P: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines both point the same way, so they are parallel. The line AB crosses both of them.Bearing of Q from P = 040°
- Find the angle at Q between the north line and QP. Angles between parallel lines and on the same side of the crossing line are co-interior and add up to 180°.180° − 040° = 140°
- Turn it into a bearing. A bearing is measured clockwise from north, but this angle is measured the other way round. Take it away from a full turn of 360°.360° − 140° = 220°
- Write it with three figures and check with the shortcut. Going back is the exact opposite way, which is half a turn (180°) different from the forward bearing.40° + 180° = 220°. Back bearing = 220°
Check. Reverse it again: 220° − 180° = 40°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of P from Q is the back bearing of the bearing of Q from P: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines both point the same way, so they are parallel. The line AB crosses both of them.Bearing of Q from P = 125°
- Find the angle at Q between the north line and QP. Angles between parallel lines and on the same side of the crossing line are co-interior and add up to 180°.180° − 125° = 55°
- Turn it into a bearing. A bearing is measured clockwise from north, but this angle is measured the other way round. Take it away from a full turn of 360°.360° − 55° = 305°
- Write it with three figures and check with the shortcut. Going back is the exact opposite way, which is half a turn (180°) different from the forward bearing.125° + 180° = 305°. Back bearing = 305°
Check. Reverse it again: 305° − 180° = 125°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of P from Q is the back bearing of the bearing of Q from P: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines are parallel. The line AB crosses both of them.Bearing of Q from P = 200°
- Find the angle between the north-south line at P and PQ. The bearing is more than 180°, so the line has gone past south. The extra angle past south is the bearing minus 180°.200° − 180° = 20°
- Use alternate angles at Q. Alternate angles (a Z shape) between parallel lines are equal. So the angle at B between the north line and BA is the same size, and it is measured clockwise from north.Angle = 20°
- Write it with three figures and check with the shortcut. Going back is half a turn different. Because the forward bearing is more than 180°, subtract 180°.200° − 180° = 20°. Back bearing = 020°
Check. Reverse it again: 20° + 180° = 200°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of P from Q is the back bearing of the bearing of Q from P: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines are parallel. The line AB crosses both of them.Bearing of Q from P = 305°
- Find the angle between the north-south line at P and PQ. The bearing is more than 180°, so the line has gone past south. The extra angle past south is the bearing minus 180°.305° − 180° = 125°
- Use alternate angles at Q. Alternate angles (a Z shape) between parallel lines are equal. So the angle at B between the north line and BA is the same size, and it is measured clockwise from north.Angle = 125°
- Write it with three figures and check with the shortcut. Going back is half a turn different. Because the forward bearing is more than 180°, subtract 180°.305° − 180° = 125°. Back bearing = 125°
Check. Reverse it again: 125° + 180° = 305°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of P from Q is the back bearing of the bearing of Q from P: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines both point the same way, so they are parallel. The line AB crosses both of them.Bearing of Q from P = 090°
- Find the angle at Q between the north line and QP. Angles between parallel lines and on the same side of the crossing line are co-interior and add up to 180°.180° − 090° = 90°
- Turn it into a bearing. A bearing is measured clockwise from north, but this angle is measured the other way round. Take it away from a full turn of 360°.360° − 90° = 270°
- Write it with three figures and check with the shortcut. Going back is the exact opposite way, which is half a turn (180°) different from the forward bearing.90° + 180° = 270°. Back bearing = 270°
Check. Reverse it again: 270° − 180° = 90°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of P from Q is the back bearing of the bearing of Q from P: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines are parallel. The line AB crosses both of them.Bearing of Q from P = 270°
- Find the angle between the north-south line at P and PQ. The bearing is more than 180°, so the line has gone past south. The extra angle past south is the bearing minus 180°.270° − 180° = 90°
- Use alternate angles at Q. Alternate angles (a Z shape) between parallel lines are equal. So the angle at B between the north line and BA is the same size, and it is measured clockwise from north.Angle = 90°
- Write it with three figures and check with the shortcut. Going back is half a turn different. Because the forward bearing is more than 180°, subtract 180°.270° − 180° = 90°. Back bearing = 090°
Check. Reverse it again: 90° + 180° = 270°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Due east is a quarter turn clockwise from north.
- Find the direction. East is at 90° from north.90°
- Write three figures. 090°
Check. East on a compass is 090°. ✓
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A is due west of B, and west is three quarter turns from north.
- Find the direction. From B, A is back to the west.West = 270°
- Check with the shortcut. 90° + 180° = 270°.270°
Check. The way back is the opposite direction. ✓
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Due south is a half turn.
- Find the direction. South is half a turn from north.180°
- Write three figures. 180 already has three figures.180°
Check. South on a compass is 180°. ✓
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A is due north of C.
- Find the direction. From C, A is straight up (north).North = 000°
- Check with the shortcut. 180° is not less than 180°, so subtract 180°: 180° − 180° = 0°.000°
Check. North is 000°, written with three figures. ✓
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at A, then measure clockwise from it to the line AB.
- Use the north line at A. The bearing of B from A starts at A.Stand at A
- Measure clockwise. Place the protractor on the point with zero on the north line. Turn clockwise to the line.The line AB is at 63.4° to 1 d.p.
- Write three figures. Round to the nearest degree.063°
Check. The line AB points into the east half of the page, which matches. ✓
Watch out. Measure from the north line at the point named after “from”.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at A, then measure clockwise from it to the line AC.
- Use the north line at A. The bearing of C from A starts at A.Stand at A
- Measure clockwise. Place the protractor on the point with zero on the north line. Turn clockwise to the line.The line AC is at 326.3° to 1 d.p.
- Write three figures. Round to the nearest degree.326°
Check. The line AC points into the west half of the page, which matches. ✓
Watch out. Measure from the north line at the point named after “from”.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at B, then measure clockwise from it to the line BC.
- Use the north line at B. The bearing of C from B starts at B.Stand at B
- Measure clockwise. Place the protractor on the point with zero on the north line. Turn clockwise to the line.The line BC is at 279.5° to 1 d.p.
- Write three figures. Round to the nearest degree.279°
Check. The line BC points into the west half of the page, which matches. ✓
Watch out. Measure from the north line at the point named after “from”.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A bearing must always have three figures.
- Say what is wrong. Bearings are written with three figures so that they cannot be confused with other angles.5° has only one figure.
- Correct it. Add two zeros at the front.005°
Check. 005° is just under 10° clockwise from north: almost due north. ✓
Watch out. Learn the pattern: 007°, 045°, 090°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of A from B is the back bearing of the bearing of B from A: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines both point the same way, so they are parallel. The line AB crosses both of them.Bearing of B from A = 130°
- Find the angle at B between the north line and BA. Angles between parallel lines and on the same side of the crossing line are co-interior and add up to 180°.180° − 130° = 50°
- Turn it into a bearing. A bearing is measured clockwise from north, but this angle is measured the other way round. Take it away from a full turn of 360°.360° − 50° = 310°
- Write it with three figures and check with the shortcut. Going back is the exact opposite way, which is half a turn (180°) different from the forward bearing.130° + 180° = 310°. Back bearing = 310°. So the bearing of A from B is 310°, as required.
Check. Reverse it again: 310° − 180° = 130°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearing of S from L is the back bearing of the bearing of L from S: add or subtract 180° so the answer stays between 0° and 360°.
- Draw a sketch with a north line at each point. The two north lines are parallel. The line AB crosses both of them.Bearing of L from S = 200°
- Find the angle between the north-south line at S and SL. The bearing is more than 180°, so the line has gone past south. The extra angle past south is the bearing minus 180°.200° − 180° = 20°
- Use alternate angles at L. Alternate angles (a Z shape) between parallel lines are equal. So the angle at B between the north line and BA is the same size, and it is measured clockwise from north.Angle = 20°
- Write it with three figures and check with the shortcut. Going back is half a turn different. Because the forward bearing is more than 180°, subtract 180°.200° − 180° = 20°. Back bearing = 020°
Check. Reverse it again: 20° + 180° = 200°. ✓
Watch out. Always draw the north line at the point you are measuring from. If the forward bearing is less than 180° add 180°; if it is more than 180° subtract 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find the back bearing of A from B. Then angle ABC is the difference between the bearings of A and C from B.
- Find the bearing of A from B. 56° is less than 180°, so add 180°.56° + 180° = 236°
- Look at the north line at B. The line BA is at 236° and the line BC is at 148°, both clockwise from north.BA = 236°, BC = 148°
- Find the difference. This is the angle between the two lines.236° − 148° = 88°
Check. The ship turned right by 148° − 56° = 92°, so the corner angle is 180° − 92° = 88°. ✓
Watch out. Use the north line at B.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. A, B and C are on one straight line, so C is in the same direction from A as B is. The bearing of A from C is the back bearing.
- Find the bearing of C from A. B is on the line AC, so the bearing of C from A is also 072°.Bearing of C from A = 072°
- Find the back bearing. 72° is less than 180°, so add 180°.72° + 180° = 252°
- Write the answer with a reason. The way back is the opposite direction (half a turn).The bearing of A from C is 252°.
Check. Draw it: the line from C back to A points south-west. 252° is in the south-west quarter. ✓
Watch out. Points on a straight line have the same bearing from the first point.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the three points. Triangle ABC has a right angle at B and two equal sides, so it is an isosceles right-angled triangle.
- Draw the diagram. B is east of A and C is south of B. East and south are at right angles.AB = BC = 6 km, angle ABC = 90°
- Find angle BAC. The two base angles of an isosceles triangle are equal and the angles add to 180°.(180° − 90°) ÷ 2 = 45°
- Find the bearing. East is 090°. C is a further 45° clockwise from east, towards south.90° + 45° = 135°
Check. 135° is exactly south-east, which is where C is. ✓
Watch out. The angle 45° is measured from the east line, not from north.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Call the bearing of B from A x. The back bearing is x + 180° (if x is less than 180°). Make an equation.
- Write the two bearings. Let the bearing of B from A be x°. The bearing of A from B is x + 180 (if x < 180).Bearing of A from B = x + 180
- Form the equation. It is three times the first bearing.x + 180 = 3x
- Solve. Subtract x from both sides, then divide by 2.180 = 2x
x = 90 - Check the condition. 90 is less than 180, so adding 180 was right.Bearing of B from A = 090°
Check. B is due east of A. The bearing of A from B is 270° = 3 × 90°. ✓
Watch out. If you assume x is more than 180° the equation is x − 180 = 3x and x comes out negative, so it cannot be right.