Topic 3 of 7 · Form 5

Interpenetration of solids and developments

The line of intersection where two solids meet (e.g. a cylinder joining a cone, two cylinders at an angle, a prism through a pyramid). Methods:

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The line of intersection where two solids meet (e.g. a cylinder joining a cone, two cylinders at an angle, a prism through a pyramid). Methods:

Generator/line method: draw lines on one surface and find where they meet the other.
Cutting plane method: take horizontal (or other) sections through both solids; their intersections give points on the curve.
Common central sphere method for intersecting cylinders and cones whose axes meet.

Parallel-line development: prisms and cylinders (including truncated and intersecting parts).
Radial-line development: pyramids and cones, using the true length of slant edges.
Triangulation: transition pieces (e.g. square-to-round ducts).

Worked example: Find the angle of the sector for the development of a cone with base radius 40 mm and slant height 120 mm.

Angle = (r/l) × 360° = (40/120) × 360° = 120°

Sheet-metal ducts, hoppers, chutes, pipe junctions, and packaging.

Exam tip: Find the critical points (highest, lowest and outermost) on an intersection curve first.

Which development method is used for a square-to-round transition piece?
A cone has radius 30 mm and slant height 90 mm. Find the sector angle of its development.
Name two methods for finding the line of intersection of two solids.

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Triangulation.
(30/90) × 360° = 120°
Any two: generator (line) method, cutting plane method, common central sphere method.

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