Technical Drawing · Lesson 7

True lengths, inclinations, traces, planes, first and second auxiliary views

WAECNECO

What the syllabus asks

WAEC: WAEC/NECO Technical Drawing, Solid Geometry item 12 (lines and planes in space: true lengths, inclinations, traces) and item 15 (auxiliary projection, first and second auxiliary views; true shapes). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

NECO: WAEC/NECO Technical Drawing, Solid Geometry item 12 (lines and planes in space: true lengths, inclinations, traces) and item 15 (auxiliary projection, first and second auxiliary views; true shapes). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

What you will learn

Project points and lines and recognise their positions from views.
Find true lengths and true inclinations by rotation, auxiliary plane and right triangle.
Find horizontal and vertical traces of lines.
Describe planes by their traces and find their true inclinations.
Draw first and second auxiliary views to show true shapes.
Apply these methods to roofs, hoppers, and inclined solids.

Key points

Elevation shows heights; plan shows depths.
View parallel to XY means the other view is true length.
TL² = Δx² + Δy² + Δz².
Rotate plan for θ; rotate elevation for φ.
HT on the HP, VT on the VP.
Auxiliary plane parallel to the face gives true shape.
Transfer distances from the grandparent view.
Oblique face: edge view first, then true shape.
True-length diagram for many slant edges.
Check drawn results by calculation.

Start here: Points and lines in space

Before we can find true lengths, true shapes or the angle between surfaces, we need a clear picture of how a single point is projected. Use the two principal planes of td5: the vertical plane (VP) and horizontal plane (HP), meeting at the XY line.

Projecting a point

A point P is fixed by three numbers: its distance along XY, its height above the HP, and its distance in front of the VP. In first-angle projection:

The elevation p′ is above XY at a distance equal to the height of P above the HP.
The plan p is below XY at a distance equal to the distance of P in front of the VP.
p and p′ always lie on the same projector, a line perpendicular to XY.

Naming convention: plan points are lower-case letters (a, b), elevation points are primed (a′, b′), and auxiliary views get subscripts or further primes (a1, a1′).

Lines

A line is fixed by its two end points. Project each end and join. Its views depend on how the line lies:

Line position	Plan	Elevation
Parallel to both planes	True length, parallel to XY	True length, parallel to XY
Perpendicular to HP (vertical)	A point	True length, ⊥ XY
Perpendicular to VP	True length, ⊥ XY	A point
Parallel to VP, inclined to HP	Parallel to XY (shortened)	True length at true angle θ to XY
Parallel to HP, inclined to VP	True length at true angle φ to XY	Parallel to XY (shortened)
Inclined to both (oblique)	Shortened	Shortened

The key insight: a line shows its true length in any view taken looking perpendicular to it, which happens only when the line is parallel to the plane of that view. When a view shows a line parallel to XY, the other view shows its true length.

A Nigerian example

A guy wire from the top of a telecom mast in Ilorin to an anchor in the ground is a line inclined to both planes if the drawing is not aligned with it. The rigger needs its true length to cut the cable; the methods in 7.2 give it.

Continue this lesson in the free app →

Practice questions

1. A line is seen in true length in the front view when it is

parallel to the VP
perpendicular to the VP
inclined to both planes
parallel to the HP only

Show answer

A: parallel to the VP
Lines parallel to a plane show true length on it.

2. A line perpendicular to the HP appears in plan as

a true-length line
a shortened line
a curve
a point

Show answer

D: a point
You look along it.

3. The point where a line, produced if necessary, meets the HP is its

true length
auxiliary point
horizontal trace
vertical trace

Show answer

C: horizontal trace
HT.

4. A line has differences of 50, 30 and 40 mm along the three axes. Its true length is about

64.0 mm
70.7 mm
120 mm
58.3 mm

Show answer

B: 70.7 mm
√(2500 + 900 + 1600) ≈ 70.7 mm.

5. The angle of that line to the HP (rise 40 mm, true length 70.7 mm) is about

34.4°
25.1°
40°
55.6°

Show answer

A: 34.4°
sin⁻¹(40/70.7) ≈ 34.4°.

Key words

Projector: Line perpendicular to XY joining views of a point.
True length: Actual length of a line.
Foreshortened: Shorter than true length in a view.
Inclination θ: Angle of a line or plane to the HP.
Inclination φ: Angle of a line or plane to the VP.
Trace: Point or line where an element meets a principal plane.
Horizontal trace: Meeting point or line with the HP.
Vertical trace: Meeting point or line with the VP.
Oblique line: Line inclined to both principal planes.
Inclined plane: Plane perpendicular to one principal plane, inclined to the other.
Oblique plane: Plane inclined to all principal planes.
Edge view: View in which a plane appears as a line.

Learn True lengths, inclinations, traces, planes, first and second auxiliary views step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android