Scale Drawing

Syllabus tag: KCSE | Mathematics | Form 1 | Topic 22 Scale Drawing

Lesson objectives

By the end of this topic, you should be able to:

• Choose and use an appropriate scale.
• Draw suitable sketches from given information.
• State the bearing of one point from another.
• Solve problems involving bearings, distances and angles of elevation and depression.

Scale Drawing

A scale drawing represents something too large to draw full size, keeping every shape and angle exactly right.

a) Choosing a scale

The scale must fit the paper and keep the drawing readable.

Find the largest measurement first, then test a scale against your page.

Choose round scales: 1 : 50, 1 : 100, 1 : 1 000, 1 : 50 000. Awkward ratios waste time.

Always write the scale on the drawing. Without it the drawing cannot be interpreted.

b) Using the scale

Going onto the drawing, divide by the scale number.

Coming off the drawing, multiply.

Convert both sides into the same unit before doing either.

c) Bearings

A bearing gives direction, measured clockwise from north, and written with three figures.

So north-east is 045°, east is 090°, south is 180° and west is 270°.

Always write three figures. Sixty-five degrees is 065°, not 65°.

d) Finding a bearing

To find the bearing of B from A, draw a north line at A. Then measure clockwise round to the line AB.

The order matters. The bearing of B from A is not the same as the bearing of A from B.

e) Back bearings

The back bearing is the reverse direction.

If the forward bearing is less than 180°, add 180°. If it is 180° or more, subtract 180°.

If B is on a bearing of 065° from A, then A is on a bearing of 245° from B.

Check: the two always differ by exactly 180°.

f) Angles of elevation and depression

The angle of elevation is measured upwards from the horizontal to the line of sight.

The angle of depression is measured downwards from the horizontal.

Both are measured from the horizontal, never from the vertical. Measuring from the vertical gives the complement, which is wrong.

The angle of elevation from A to B equals the angle of depression from B to A. The two horizontals are parallel, so these are alternate angles.

g) Solving by scale drawing

Draw the situation to scale, using a ruler and a protractor.

Measure the unknown distance or angle on your drawing.

Convert measured lengths back using the scale.

Accuracy depends on the drawing. Use a sharp pencil and read carefully.

h) Where this is used

Surveying land. Navigation at sea and in the air. Architectural plans. Finding the height of a tree or building without climbing it.

Words to know

• Scale -- the ratio between a drawing length and the corresponding real length.
• Bearing -- a direction measured clockwise from north, written in three digits.
• Back bearing -- the bearing of the reverse direction, differing by 180°.
• Angle of elevation -- the angle above the horizontal when looking up.
• Angle of depression -- the angle below the horizontal when looking down.

:::checkpoint Check yourself

1. Write the bearing of due south in three figures.
2. On a 1 : 25 000 map, two points are 8 cm apart. Find the actual distance in km.
3. A point B is on a bearing of 120° from A. What is the bearing of A from B?
4. From where is an angle of depression measured? :::

Bridge to practice

The exercises begin with scales and conversion, move through bearings and back bearings, and finish with elevation, depression and full navigation problems. For every bearing question, write down which point you are standing at before measuring anything.