Technical Drawing · Lesson 9

Parallel-line, radial-line and triangulation developments, transition pieces

WAECNECO

What the syllabus asks

WAEC: WAEC/NECO Technical Drawing, Solid Geometry item 17 (development of surfaces of prisms, cylinders, pyramids and cones, including cut solids and transition pieces). (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 17 (development of surfaces of prisms, cylinders, pyramids and cones, including cut solids and transition pieces). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

What you will learn

Distinguish developable and non-developable surfaces and choose the method.
Develop right and cut prisms and cylinders by the parallel-line method.
Develop right and cut pyramids and cones by the radial-line method.
Develop oblique cones and transition pieces by triangulation.
Develop the parts of intersecting solids and approximate spheres.
Apply practical sheet-metal allowances and checks.

Key points

Every line in a development is true length.
Prisms and cylinders: parallel-line.
Pyramids and cones: radial-line.
Transition pieces: triangulation.
Girth πd; calculate rather than step chords.
Cone sector angle 360r/L.
Pyramid radius is the slant edge.
Project cut points horizontally to the true-length outline.
Seam on the shortest line.
Check by folding a paper model.

Start here: What development means

A development (or net) is the shape of a surface laid out flat. Fold or roll it up and you get the object. Sheet-metal workers at Owode Onirin or Ladipo use developments every day to cut zinc, aluminium and galvanised sheet for gutters, funnels, buckets, chimney hoods, ducts, hoppers and the trays of grain dryers. Packaging designers develop cartons for juice and noodles. In the exam, WAEC solid geometry item 17 asks for developments of prisms, cylinders, pyramids and cones, cut or uncut, and of transition pieces.

Developable and non-developable surfaces

Developable: surfaces that can be unrolled flat without stretching: planes, prisms, pyramids, cylinders and cones.
Non-developable: doubly curved surfaces such as spheres and domes; they can only be approximated (by gores or zones, as in the panels of a football).

Three methods

Method	Used for
Parallel-line	Prisms and cylinders (edges or generators are parallel)
Radial-line	Pyramids and cones (edges or generators meet at the apex)
Triangulation	Transition pieces and oblique surfaces

The golden rule

Every line in a development must be a true length. The edges of a prism standing upright are already true length in elevation, and the base perimeter is true in plan. For pyramids and oblique surfaces you must first find true lengths (td7). Most errors in development come from using a foreshortened length from a view.

Seams and allowances

In practice, a seam (where the development joins) is placed along the shortest edge to save welding or riveting, and extra material is added for laps, folded seams and wired edges. Exam developments are drawn without allowances unless asked, but you should know they exist. A grooved seam on a 0.5 mm galvanised bucket typically needs about three times the groove width in extra metal (workshop rule of thumb).

Worked example 9.F (choosing the method)

Name the method to develop: (a) a square duct cut at 30°; (b) a conical hopper; (c) a rectangle-to-round transition; (d) an offset cone.

(a) Parallel-line (edges parallel).
(b) Radial-line (generators meet at the apex).
(c) Triangulation (no common apex, no parallel edges).
(d) Triangulation, or radial-line using individually found true lengths.

Parallel-line; radial-line; triangulation; triangulation.

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Practice questions

1. A cylinder is developed by the

parallel-line method
radial-line method
triangulation method
common-sphere method

Show answer

A: parallel-line method
Generators are parallel.

2. A cone is developed by the

parallel-line method
auxiliary method
trammel method
radial-line method

Show answer

D: radial-line method
Generators radiate from the apex.

3. A square-to-round transition piece is developed by

the radial-line method from one apex
oblique projection
triangulation
the parallel-line method

Show answer

C: triangulation
No common apex and no parallel edges.

4. The development of a right cone of base radius 30 mm and slant height 76.2 mm is a sector of angle about

76.2°
141.8°
90°
180°

Show answer

B: 141.8°
360 × 30/76.2 ≈ 141.8°.

5. A cylinder of diameter 50 mm develops into a rectangle of length about

157.1 mm
78.5 mm
100 mm
314.2 mm

Show answer

A: 157.1 mm
πd.

Key words

Development: Flat pattern of a surface.
Net: Another name for a development.
Developable surface: Surface that unrolls flat without stretching.
Stretch-out line: Base line of a parallel-line development.
Girth: Perimeter or circumference to be developed.
Generator: Straight line on a cone or cylinder surface.
Slant height: Length of a cone generator.
Slant edge: Edge from apex to base corner of a pyramid.
Sector: Part of a circle between two radii.
Frustum: Cone or pyramid with the top cut off parallel to the base.
Annular sector: Sector of a ring.
Oblique cone: Cone with the apex not over the base centre.

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