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Cambridge IGCSE0580

Scale drawings

Mathematics 0580 Chapter Notes

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Scale drawings
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Scale drawings notes

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In this chapter

  1. 1Scales and real distances
  2. 2Drawing and measuring accurately
  3. 3Three-figure bearings and compass points
  4. 4Measuring and drawing a bearing
  5. 5Back bearings

1. Scales and real distances

A scale links a length on a drawing to the real length. "1 cm represents 200 m" means every centimetre on the drawing stands for 200 m.

To find a real distance, multiply the drawing length by the scale. To find a drawing length, divide the real length by the scale. Put both lengths in the same units before you write the scale as 1 : n.

real length = drawing length × scale

1 cm to 200 m = 1 : 20 000

Examiner insight

“A common sense approach would have shown that an answer such as 3 900 000 km apart was not a practical answer.”

Worked example 12 marks

A scale drawing shows the positions of two schools, A and B. The scale is 1 centimetre represents 200 metres.

(b) Write the scale in the form 1 : n.

(c) On a second scale drawing, A and B are four times as far apart as on the first scale drawing. Write down the scale of the second scale drawing. Give your answer in the form 1 : n.

  1. 1

    (b) 200 m = 20 000 cm, so the scale is 1 : 20 000 [1]

  2. 2

    (c) Four times as far apart means each centimetre stands for a quarter of the distance: 20 000 ÷ 4 = 5000, so 1 : 5000 [1]

2. Drawing and measuring accurately

A ruler must be used for all straight edges of a scale drawing.

Work out each drawing length first, then draw it. Measure with care: a mark scheme accepts only a narrow range, such as 18.4 to 19.2 km for a distance near 19 km. Changing the scale changes the distances on the drawing, not the bearings.

Examiner insight

“Many candidates did not appear to appreciate that the bearing would remain the same and that the distance would change with the different scale used.”

Worked example 13 marks

Do not use a calculator. A path is 4.2 km long. It is to be drawn on a scale of 1 cm to 600 m. Find the length of the path on the drawing.

  1. 1

    4.2 km = 4200 m [1]

  2. 2

    drawing length = 4200 ÷ 600 [1]

  3. 3

    = 7 cm [1]

3. Three-figure bearings and compass points

A bearing is an angle measured clockwise from north. It always has three figures, from 000° to 360°. Write 25° as 025°.

The four compass points are north, east, south and west. Point D is due east of point C when D lies directly to the east of C.

N = 000°, E = 090°, S = 180°, W = 270°

Examiner insight

“Some answers were written as a 2-figure angle rather than a 3-figure bearing.”

Worked example 12 marks

Do not use a calculator. Point D is due east of point C. Point E is due south of point C. Write down the bearing of D from C and the bearing of E from C.

  1. 1

    east is a quarter turn clockwise from north: the bearing of D from C is 090° [1]

  2. 2

    south is a half turn clockwise from north: the bearing of E from C is 180° [1]

4. Measuring and drawing a bearing

To find the bearing of B from A, stand at A. Draw a north line through A. Measure the angle clockwise from the north line to the line AB.

To mark a point, measure the angle clockwise from north at the starting point. Then measure the drawing distance along that line.

Read the protractor scale that starts at zero on the north line. Then check the size of your answer: a point that lies between east and south of A has a bearing between 090° and 180°.

Examiner insight

“The most successful candidates began their solution by drawing a diagram to illustrate the given bearing.” “Candidates continue to find measuring bearings very challenging.”

Worked example 14 marks

A village, V, is 94 km from a town, T, on a bearing of 112°. The scale is 1 centimetre represents 20 kilometres. Describe how to mark the position of V on a scale drawing.

  1. 1

    drawing length = 94 ÷ 20 = 4.7 cm [1]

  2. 2

    put the centre of the protractor on T, with its zero on the north line [1]

  3. 3

    measure 112° clockwise from north and mark it [1]

  4. 4

    draw a line from T through the mark and mark V at 4.7 cm from T [1]

5. Back bearings

The bearing of A from B is the reverse of the bearing of B from A. The two bearings differ by 180°. Candidates who draw a sketch diagram tend to be more successful, so draw one with a north line at each point before you calculate.

If the first bearing is less than 180°, add 180°. If it is 180° or more, subtract 180°. The answer must stay between 000° and 360°.

bearing < 180°: back bearing = bearing + 180°

bearing ≥ 180°: back bearing = bearing - 180°

Examiner insight

“Many candidates did not appear to appreciate the calculation required was either 180 + 73 or 360 – 107.” “Common errors included 360 – 73 = 287, 180 – 73 = 107 and 90 + 73 = 163, with a significant number of nil responses.”

Worked example 12 marks

Do not use a calculator. The bearing of B from A is 025°. Find the bearing of A from B.

  1. 1

    025° is less than 180°, so add 180°: 25 + 180 [1]

  2. 2

    the bearing of A from B is 205° [1]

Recap

  • Real length = drawing length × scale. Drawing length = real length ÷ scale.
  • Use the same units before writing a scale as 1 : n.
  • Use a ruler for every straight edge. A new scale changes distances, not bearings.
  • A bearing is measured clockwise from north and has three figures.
  • North 000°, east 090°, south 180°, west 270°.
  • Back bearing: add 180° if under 180°, subtract 180° if 180° or more.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A map has a scale of 1 cm to 250 m. Two villages are 6 cm apart on the map. Find the real distance between them in kilometres.3 marks
  2. Do not use a calculator. Write the scale 1 cm represents 50 m in the form 1 : n.2 marks
  3. Do not use a calculator. A map has a scale of 1 : 25 000. Two points are 3 km apart. Find the distance between them on the map in centimetres.3 marks
  4. Point D is due west of point C. Write down the bearing of D from C and the bearing of C from D.2 marks
  5. Do not use a calculator. The bearing of town B from town A is 064°. Find the bearing of A from B.2 marks
  6. A scale drawing uses 1 cm to 4 km. A lighthouse L is 6.5 cm from a harbour H on the drawing. L is on a bearing of 040° from H. (a) Work out the real distance HL. (b) Find the bearing of H from L.3 marks
  7. Do not use a calculator. The bearing of Q from P is 312°. Find the bearing of P from Q.2 marks
  8. A field has a length of 8 cm on a plan. Its real length is 200 m. (a) Write the scale of the plan in the form 1 : n. (b) A second plan has scale 1 : 5000. Find the length of the field on the second plan.3 marks
  9. Do not use a calculator. A scale drawing has a scale of 1 : 50 000. On it, a mast M is 7.2 cm from a farm F, and M is on a bearing of 236° from F. The drawing is made again with a scale of 1 : 20 000. (a) Find the real distance FM in kilometres. (b) Find the length of FM on the new drawing. (c) Find the bearing of F from M on the new drawing.5 marks
  10. A plan of a field has a scale of 1 : 400. A straight fence runs from post P to post Q. On the plan, PQ is 13.5 cm long and Q is on a bearing of 305° from P. A gate G is on the fence, 18 m from Q. (a) Find the real length of the fence in metres. (b) Find the distance of G from P on the plan. (c) Find the bearing of G from Q. (d) On a second plan the fence is 21.6 cm long. Write the scale of the second plan in the form 1 : n.6 marks

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