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Oblique cone flat pattern calculator

A full cone with its apex off-centre over a round base. Enter the base diameter, the apex height and the offset; you get the pattern laid out around the apex.

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Enter your measurements

Your dimensions are measured to the

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1:1 PDF and DXF files of this flat pattern are made in Trueflat, ready for print, plasma, laser or waterjet. It works offline on site.

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621.3 mm452.2 mm
Flat pattern: cut on the bold outline, bend on the dashed blue lines, the thin lines are marking lines

Seam

812.1mm

Approximate, within ±0.1 mm

Slant length
In the app
Arc length
In the app
Sheet size
In the app

Every result is in Trueflat

The free web version shows the main result and the drawing. Trueflat gives you every mark and measurement, and exports a cut sheet. It is free to download.

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How to lay it out

  1. Divide the base

    Draw the base circle in plan with the apex point beside or inside it, and divide the circle into equal parts.

  2. Find the true lengths

    For each division, take its plan distance to the apex and the apex height as the legs of a right triangle. The hypotenuse is the element's true length.

  3. Lay out from the apex

    Draw the shortest element from a centre point. Swing the next element length from the centre and the base chord from the last point; the crossing is the next point. Repeat around the base.

  4. Draw the base edge

    Bend a batten through the points, join the last point back to the centre for the second seam line and add your allowance.

Good to know

  • This pattern is approximate; the result card shows how close it is.
  • Source: Kittredge, The New Metal Worker Pattern Book (1917), Problem 163 "The Envelope of a Scalene Cone", Figs. 526–529, p. 307

Method and formula

In an oblique (scalene) cone the apex is not over the centre of the base, so the elements have different lengths. Each one keeps its true length from the apex on the pattern; the base edge is a closed curve rather than a circular arc. The surface unrolls exactly.

  • Element length at plan angle θ from the short side: √((r − X)² + H² + 4rX sin²(θ ÷ 2)).
  • Shortest √((r − X)² + H²), longest √((r + X)² + H²).
  • Base edge length on the pattern is π × D, the base circumference.

Stepping chords by hand, as in the steps, gives a pattern that is slightly short. Trueflat integrates the angle between elements numerically, so the points lie on the true curve. With an offset of 0 the result is a right cone. The apex may also lie outside the base circle. Thickness is taken to the middle of the sheet only.

Worked example

D = 300 mm, H = 400 mm, X = 100 mm: shortest element √(50² + 400²) = 403.11 mm, longest √(250² + 400²) = 471.70 mm. The pattern spans 122.01°, and the base edge is 942.48 mm long, which is π × 300.

Questions

Where do I measure the offset?

Horizontally, from the centre of the base circle to the point directly under the apex. With an offset equal to the base radius, one element is vertical.

How is this different from a right cone?

A right cone's pattern is a sector of a circle, set by one radius and one angle. An oblique cone's pattern has a different radius along every element, so you need a true length for each division.

Is the pattern exact?

The surface is a cone, so it unrolls without stretching. There is no closed formula for the angles, so Trueflat integrates them numerically, and the outline is drawn as straight chords through the curve samples. More samples bring the drawn edge closer to the true curve.

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