How to lay out a cone flat pattern
To lay out a cone flat pattern, find the slant height from the height and the difference in the end radii. From one center, swing the outside radius (large radius × slant ÷ radius difference) and the inside radius (small radius × slant ÷ radius difference), step off the arc angle between two lines from the center, and cut.
Drawn by Stretchout from the worked example below. Radial line development.
What you need
- Plate or sheet big enough for the blank
- A trammel (beam compass) for the long radius
- Dividers for stepping off the arc
- A tape, a straightedge and soapstone or a scribe
Worked example
We’ll lay out a 24" to 12" OD concentric cone, 18" high, in 3/16" plate, laid out on the mean line. The numbers:
- Mean radii, large and small
- 11 15/16" and 5 15/16"
- Slant height
- 19"
- Outside radius
- 37 5/8"
- Inside radius
- 18 11/16"
- Arc angle
- 113.84°
- Blank
- 63 1/8" × 27 7/16"
Step by step
Step 1
Mark the center. Swing the outside radius, 37 5/8". That arc is the large end.
Step 2
From the same center, swing the inside radius, 18 11/16". That arc is the small end.
Step 3
Draw the first seam from the center out through both arcs. Call it line 1.
Step 4
From where line 1 crosses the outside arc, measure 63 1/8" straight across to the arc (the chord of 113.84° of arc). Draw the second seam from the center through that mark.
Step 5
Step the layout lines off 4 11/16" apart on the outside arc (16 around the whole cone) and draw each one toward the center.
Step 6
Cut on the two arcs and the two seams.
Watch it laid out
Every step of the worked example. Play, or step through with Next and Back (or the arrow keys).
1.Mark the center. Swing the outside radius, 37 5/8". That arc is the large end.
The formulas
Take R1 and R2 as the large and small radii and H as the height between the ends. The slant height is s = √(H² + (R1 − R2)²).
Outside radius = R1 × s ÷ (R1 − R2). Inside radius = R2 × s ÷ (R1 − R2). Arc angle = 360° × (R1 − R2) ÷ s. A cone run to a point is the same with R2 = 0: no inside radius.
Use radii on the line you lay out on. For plate that rolls, that’s the mean (mid-thickness) line: the OD radius less half the thickness. Laying out on the OD makes the cone roll about one plate thickness too big across.
Questions
What is a truncated cone flat pattern?
A cone with its top cut off: two round ends, like a concentric reducer. Its flat pattern is part of a ring: two arcs from one center, joined by two straight seams.
Which diameter do I lay out on?
The mean line (mid-thickness) for plate that you roll. Thin sheet can go on either face. The calculator lays out on OD, ID or mean and tells you which it used.
What if the small end is off center (an eccentric reducer)?
Radial line only works when the ends are centered. Off center, the cone is laid out by triangulation, one triangle at a time. Switch on Eccentric or Offset in the calculator and it walks the triangulation step by step.
Why did my triangulated cone come out short?
If you space the points with the chord measured on the round, each space is a little short on the flat, and it adds up: about half an inch around a 24" cone at 16 spaces. Take the spacing on the sheet instead; the step-by-step in the calculator gives it.
Can I make it in more than one piece?
Yes. Split the arc angle evenly: each piece gets the same two radii and its share of the angle. The calculator splits into 1 to 8 pieces and flags a blank that won’t fit a 60" × 120" sheet.
What about the weld gap?
Move each seam line in half the gap, square to it, so the cone still rolls to size with the gap. The calculator does it when you enter a gap.
Your sizes
Below is the cone calculator, opened on the worked example. Put in your own sizes and it redoes every number and every step for your fitting. It’s also at /stretchout/shapes/cone.