Technical Drawing · Lesson 3

Plain, diagonal and chord scales, similar figures, enlargement by area, tangency and blending arcs

WAECNECO

What the syllabus asks

WAEC: WAEC/NECO Technical Drawing, Plane Geometry items 4 (plain and diagonal scales, scale of chords), 8 (tangency: tangents, blending arcs, spanner and anchor outlines) and 10 (similar figures; enlargement and reduction by side and by area). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

NECO: WAEC/NECO Technical Drawing, Plane Geometry items 4 (plain and diagonal scales, scale of chords), 8 (tangency: tangents, blending arcs, spanner and anchor outlines) and 10 (similar figures; enlargement and reduction by side and by area). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

What you will learn

Calculate and state representative fractions and convert between drawing and real lengths.
Construct plain and diagonal scales and read lengths from them.
Construct and use a scale of chords.
Draw similar figures by radial, parallel and grid methods.
Enlarge or reduce figures by a given side, ratio or area.
Construct tangents and blending arcs and outline objects such as spanners and anchors.

Key points

RF = drawing ÷ actual, same units.
Dimension true sizes, never scaled lengths.
Plain scale: zero after the first, subdivided division.
Diagonal scale: ten rows give tenths of a subdivision.
Chord = 2r sin(θ/2); chord 60° = radius.
Similar figures: length k, area k², volume k³.
Area change by m/n: side × √(m/n), by mean proportional.
Tangent ⊥ radius at contact.
External blend R + r; internal R − r.
Mark tangent points; line in arcs before straights.

Start here: Why drawings are made to scale

Very few objects can be drawn full size on an A2 sheet. A three-bedroom bungalow in Gwarinpa may be 15 m long; a wristwatch gear may be 3 mm across. A scale is the fixed ratio between a length on the drawing and the corresponding length on the real object. Every measurement on the drawing obeys the same ratio, so the drawing remains a faithful, measurable picture of the object.

Three kinds of scale

Full size, 1 : 1. Drawing length equals real length. Used for small machine parts.
Reduction, such as 1 : 2, 1 : 5, 1 : 10, 1 : 20, 1 : 50, 1 : 100. The drawing is smaller. Used for buildings, furniture, large machines and maps.
Enlargement, such as 2 : 1, 5 : 1, 10 : 1. The drawing is larger. Used for small parts such as watch components, electronic connectors and fine threads.

The first number always refers to the drawing and the second to the object. So 1 : 50 means “1 unit on paper represents 50 of the same units on the object”.

Representative fraction (RF)

The representative fraction is the scale written as a fraction with both lengths in the same units:

RF = length on drawing ÷ actual length

RF less than 1: reduction. RF equal to 1: full size. RF greater than 1: enlargement.
The units must agree. If 1 cm on the drawing represents 5 m, convert 5 m to 500 cm, so RF = 1/500.
If 2 cm represents 1 km, convert 1 km = 100 000 cm, so RF = 2/100 000 = 1/50 000.

Recommended scales

The international convention (ISO 5455) recommends multiples of 1, 2 and 5 times powers of ten: 1 : 2, 1 : 5, 1 : 10, 1 : 20, 1 : 50, 1 : 100, 1 : 200, 1 : 500, 1 : 1000, and enlargements 2 : 1, 5 : 1, 10 : 1, 20 : 1, 50 : 1. Architects in Nigeria commonly use 1 : 100 for floor plans and elevations, 1 : 50 for larger plans, 1 : 20 or 1 : 10 for details and 1 : 500 or 1 : 1250 for site plans.

Writing the scale on the drawing

State the main scale in the title block, for example SCALE 1 : 50. If one view uses a different scale, write it under that view. A dimension is always the true size of the object, never the measured length on the paper: a 4 m wall drawn at 1 : 100 is 40 mm long on paper but is dimensioned 4000 (mm).

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Practice questions

1. A drawing is made at 1 : 50. A wall 6 m long is drawn

120 mm long
300 mm long
12 mm long
60 mm long

Show answer

A: 120 mm long
6000 ÷ 50 = 120 mm.

2. On a plan, 5 km is shown by 25 cm. The representative fraction is

1 : 2 000
1 : 200 000
1 : 5 000
1 : 20 000

Show answer

D: 1 : 20 000
0.25 m ÷ 5000 m = 1/20 000.

3. A scale that can read three units (for example metres, decimetres and centimetres) is a

scale of chords
vernier protractor
diagonal scale
plain scale

Show answer

C: diagonal scale
Diagonals give the third unit.

4. On a 1 : 50 diagonal scale, 3.47 m measures

6.94 mm
69.4 mm
34.7 mm
173.5 mm

Show answer

B: 69.4 mm
3.47 × 20 mm = 69.4 mm.

5. On a scale of chords of radius 60 mm, the chord for 60° is

60 mm
30 mm
52 mm
120 mm

Show answer

A: 60 mm
The 60° chord equals the radius.

Key words

Scale: Fixed ratio between drawing length and real length.
Representative fraction: Scale as a fraction in the same units.
Reduction scale: Drawing smaller than the object.
Enlargement scale: Drawing larger than the object.
Plain scale: Drawn line reading two units.
Diagonal scale: Scale reading a third unit with diagonals and rows.
Scale of chords: Line of chord lengths used to set out angles.
Chord: Straight line joining two points on a circle.
Similar figures: Same shape, sides in one ratio, equal angles.
Centre of similarity: Point from which an enlargement is projected.
Scale factor: Ratio of new length to old.
Mean proportional: x where x² = ab.

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