Bearings and Scale
A bearing describes a direction as an angle measured clockwise from north, always written using three figures, for example 060 degrees or 215 degrees, and its back bearing, the return direction, is found by adding 180 degrees if the original bearing is less than 180, or subtracting 180 if it is 180 or more. A scale, such as 1 cm to 5 km, represents real distances on a map or drawing, so an actual distance is found by multiplying the drawn length by the scale.
Before you start
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Method
- Always measure a bearing clockwise from the north line at the starting point, and write it using three figures, adding leading zeros if needed, for example 7 degrees is written 007.
- To find a back bearing, add 180 degrees if the given bearing is less than 180, or subtract 180 degrees if it is 180 or more.
- Angle facts for parallel lines (the north lines at two different points are parallel) often connect a bearing and its back bearing, using alternate angles or angles on a straight line.
- For a scale drawing or map, read the scale carefully, for example 1 cm represents 5 km, then multiply a measured length by the scale to find the real distance.
- To go the other way, divide a real distance by the scale to find the length to draw on the scale drawing.
- Keep units consistent throughout a calculation, converting between cm, m and km as needed before multiplying or dividing by the scale.
Worked example
A hiker walks from a campsite on a bearing of 072 degrees to reach a waterfall. Find the bearing she would need to walk on to return directly from the waterfall to the campsite (the back bearing).
- The bearing from the campsite to the waterfall is 072 degrees, which is less than 180 degrees.
- To find the back bearing, add 180 degrees: 072 + 180 = 252.
- The bearing from the waterfall back to the campsite is 252 degrees.
Practice questions
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Q1A boat sails from a harbour on a bearing of 048 degrees. Find the back bearing, from the boat's position back to the harbour.Show answer
Answer: 048 + 180 = 228 degrees
Q2A plane flies from an airport on a bearing of 205 degrees. Find the back bearing.Show answer
Answer: 205 - 180 = 025 degrees
Q3Write the bearing 8 degrees using the correct three-figure notation.Show answer
Answer: 008 degrees
Q4A scale drawing has a scale of 1 cm to 4 km. A road on the drawing measures 7 cm. Find the real length of the road.Show answer
Answer: 7 x 4 = 28 km
Q5A map has a scale of 1 cm to 2.5 km. Two campsites are 30 km apart in real life. Find the distance between them on the map.Show answer
Answer: 30 / 2.5 = 12 cm
Q6A scale drawing uses a scale of 1 : 20000. A path measures 6 cm on the drawing. Find the real length of the path, in metres.Show answer
Answer: 1 cm represents 20000 cm = 200 m, so 6 cm represents 6 x 200 = 1200 m
Q7From a lighthouse, a buoy is on a bearing of 310 degrees. A student says the bearing from the buoy back to the lighthouse is 130 degrees. Show that the student is correct.Show answer
Answer: 310 is greater than or equal to 180, so subtract 180: 310 - 180 = 130 degrees, so the student is correct
Q8A cycle route is drawn to a scale of 1 cm to 3.5 km. The real route is 21 km long. Find the length of the route on the scale drawing.Show answer
Answer: 21 / 3.5 = 6 cm
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Village A and village B are shown on a map. The bearing of B from A is 130 degrees. (a) Find the bearing of A from B (the back bearing). (b) The map has a scale of 1 cm to 8 km, and villages A and B are 4.5 cm apart on the map. Find the real distance between them.
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A ship leaves a port and sails on a bearing of 065 degrees for 60 km to reach a buoy. (a) Find the bearing the ship would need to sail on to return directly from the buoy to the port. (b) The ship's route is shown on a chart with a scale of 1 cm to 15 km. Find the length, in cm, used to draw the ship's outward route on the chart. (c) A second ship leaves the same port on the same bearing, 065 degrees, but stops after sailing only 24 km. Explain how you know that this ship's position lies exactly on the straight-line route from the port to the buoy.
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Free printable worksheet
Want more practice on paper? Download the bearings and scale worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 19 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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