Cone and truncated cone development
Enter two diameters, the height and the plate thickness — the calculator gives the two radii and the sector angle to mark out, draws the blank, lists the points for a tape measure and exports a DXF for the cutting table. Small patterns print at full size.
Marking-out points, mm
Measured from the top left corner of the blank: X to the right, Y downwards. The points are spaced at equal fractions of the angle, thirteen to each arc.
| No. | Large arc: X | Y | Small arc: X | Y |
|---|
How to mark out the pattern
- Find the centre. Punch a centre mark on the sheet, or on a strip of scrap clamped beside it if the radius is longer than the sheet.
- Strike two arcs. Radius R1 for the large end and R2 for the small end, both from the same centre.
- Set the angle. Step the chord of the large arc along the R1 arc: its two ends are the ends of the pattern. Draw the straight edges from those points towards the centre.
- Check before cutting. The length of the large arc must equal π × the mean large diameter. Run a flexible tape along the arc — it is the quickest way to catch a slipped trammel.
What is calculated
The side of a cone unrolls into a sector of a ring exactly, with no stretching — just as a cylinder unrolls into a rectangle. For a truncated cone (frustum) with diameters D and d and height H:
- slant height s = √(H² + ((D − d)/2)²);
- large radius R1 = s·D / (D − d), small radius R2 = R1 − s;
- sector angle α = 180°·(D − d) / s.
Plate has thickness, so the diameters used are those of the middle surface: each outside diameter is reduced by t·cos θ, where θ is the half-angle of the cone. On 2 mm plate the difference is small; on 10 mm it decides whether the cone closes.
Worked example
A reducer from 400 to 200 mm outside, 300 mm high, in 2 mm plate. The slant height is √(300² + 100²) = 316.2 mm. On the middle surface the diameters are 398.1 and 198.1 mm, which gives R1 = 629.5 mm, R2 = 313.2 mm and a sector of 113.84°. The chord of the large arc is 1054.9 mm, the blank is 1055 × 458 mm and the part weighs 4.65 kg in steel.
Ignore the thickness and the radii come out as 632.5 and 316.2 mm: the large arc is 6 mm too long, and the cone ends up 1.9 mm oversize on the diameter.
A full cone — a cowl 300 mm across and 150 mm high in 1 mm sheet — is a single sector with R1 = 211.6 mm and an angle of 254.56°.
What the pattern does not know
- Seam allowance. The straight edges meet with no gap and no lap.
- Flat ends after rolling. Rolls leave a straight strip at each edge of the plate. Pre-bend the edges or leave extra and trim it after rolling.
- Oblique cones. Only a right cone with a circular base is covered — an eccentric reducer follows a different pattern.
The mass of a flat blank of any size is worked out by the sheet metal weight calculator. For a branch between two pipes use the pipe notching template.
Frequently asked questions
What is the formula for the flat pattern of a truncated cone?
Three figures define it. The slant height s = √(H² + ((D − d)/2)²). The large radius R1 = s·D/(D − d) and the small radius R2 = R1 − s. The sector angle α = 180°·(D − d)/s. D and d are the diameters of the middle surface of the plate, H is the height of the cone.
Which diameter goes into the pattern — outside, inside or mean?
The mean one. When plate is rolled the outside fibres stretch and the inside ones compress; only the middle surface keeps its length. Enter the outside diameters and the thickness, and the calculator moves both diameters inwards by t·cos θ, where θ is the half-angle of the cone. With the thickness set to 0 the pattern is built on the diameters exactly as entered.
How do I mark out the pattern without a protractor?
With the chord. Strike the large arc with radius R1 from the centre, pick a starting point on it and step off the “chord of the large arc” with a tape or trammel — the second point is the other end of the pattern. Join both points to the centre and strike the small arc with R2. The chord works while the sector is under 180°; for a wider one split the pattern into two pieces.
The radius is longer than my sheet. How do I mark it out?
Use the table of points under the calculator: thirteen points on each arc, given as X and Y from the corner of the blank. Mark them with a square and a tape and join them with a flexible batten. It is the usual method for shallow cones, where the centre lies metres away from the plate.
When should the cone be made in several pieces?
When the blank does not fit the sheet, when the sector is wider than about 180° and nests badly, or when the rolls cannot take the full width. Set the number of pieces: each one gets the same radii and the angle α divided by that number. Every extra piece adds one seam along the slant height.
Does the pattern include an allowance for the seam?
No. The edges of the pattern meet exactly, which is right for a butt weld with no gap on thin plate. For a root gap, take half the gap off each straight edge; for a lap or a locked seam add the lap on one edge yourself.
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