Square-to-Round Transition: Development by Triangulation
A sheet-metal shop is making a ventilation transition: a horizontal square opening 100 mm × 100 mm at the top, a horizontal circular opening Ø50 mm at the bottom, concentric with it, with a vertical height of 80 mm between the two openings. It is to be folded from 0.8 mm galvanised sheet in one piece with a folded seam, and the foreman needs the flat development marked out on the sheet before cutting. An apprentice suggests treating the job like a cylinder and using parallel-line development, rolling the pattern around the 50 mm circle. The foreman rejects this and asks for a triangulated development, with the seam placed on one of the flat sides. Only a set square, dividers and a scale rule are available, and the pattern must fit on a 500 mm × 400 mm sheet.
1. Explain why parallel-line development fails here and how the surface must be divided instead.
Read the worked answer
Parallel-line development works only when every generator of the surface is parallel to the others - a cylinder, a prism, or a pipe cut obliquely. The developed width is then a constant circumference or perimeter, measured once and stepped off along a single straight edge.
In a square-to-round transition the generators converge: the top edge is a straight 100 mm side, the bottom is a 50 mm diameter arc, and the surface between them is a set of flat triangles and conical corner patches whose slant lengths change from point to point. Rolling the pattern around the circle would put the wrong length on every seam and the piece would not close.
Correct division:
- Divide the Ø50 circle into 12 equal parts (30° steps), giving points at 0°, 30°, 45°, 60°, 90° and so on.
- Treat each of the four 100 mm sides of the square, with the circle point facing its mid-point, as a flat isosceles triangle.
- Treat each square corner with the two adjacent circle points, split by the 45° point, as two corner triangles. Total: 4 flat triangles plus 8 corner triangles = 12 triangles, each developed from true lengths.
2. Calculate the true lengths needed for one corner region and one flat triangle.
Read the worked answer
Take the top corner A at (50, 50, 80) with the axis as origin and the bottom circle in the z = 0 plane. True length = √(horizontal distance² + 80²).
Corner A to the circle point at 0°, P0 = (25, 0, 0): Δx = 25, Δy = 50, Δz = 80 → √(625 + 2500 + 6400) = √9525 = 97.6 mm.
Corner A to the circle point at 45°, P45 = (17.68, 17.68, 0): Δx = 32.32, Δy = 32.32, Δz = 80 → √(1044.6 + 1044.6 + 6400) = √8489 = 92.1 mm.
Chord P0–P45 = 2 × 25 × sin 22.5° = 19.1 mm (true in the bottom view, and true on the development because it lies in a horizontal plane).
Flat triangle: base = one 100 mm side of the square (true length); the two sides are corner-to-P0 distances = 97.6 mm each; its height on the development is √(97.6² − 50²) = 83.8 mm.
Do not use the elevation lengths directly for the inclined seams - the elevation foreshortens them to 80 mm and less.
3. Describe how the pattern is set out on the sheet, where the seam goes, and what allowance must be added.
Read the worked answer
- Draw the flat triangle first: base 100 mm, sides 97.6 mm, apex at P0. Its base is a fold line (the top edge of the square).
- On each side of the triangle build the corner triangles: from the corner set dividers to 92.1 mm and from P0 to 19.1 mm; the intersection is P45. Continue from P45 with the next chord 19.1 mm and the next corner radius 92.1 mm (or 97.6 mm at the next corner), working round the pattern.
- Alternate flat and corner triangles until all 12 are set out; the pattern should close exactly. Any gap means an arithmetic or stepping error and must be found before cutting, not corrected by forcing the last seam.
- Put the seam on one flat side: the flat triangles are plain planes, so the seam folds on a straight line and the joint sits on a flat face of the duct, which is easier to fold and to rivet than a corner patch.
- Add the folded seam allowance plus bend allowance for the 0.8 mm sheet at every fold (typically one material thickness per bend plus the lap), and check the overall pattern size: roughly 12 chords of 19.1 mm plus the flat bases, which fits comfortably on the 500 × 400 sheet once allowance is added.