FabLayout

Cone cut parallel to its axis flat pattern calculator

A cone with one side sliced off by a plane parallel to its axis, as when a cone sits against a wall. You get two pieces: the cone with its cut-out and the flat patch that closes it.

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769.1 mm492.2 mm
Flat pattern: cut on the bold outline, bend on the dashed blue lines, the thin lines are marking lines

Seam

1,479.9mm

Exact geometry

Cut length
In the app
Slant 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. Swing the full cone

    Swing an arc at the slant length from a centre point and mark the sector angle, as for a plain cone.

  2. Find where the cut starts

    On both seam lines, mark the point where the plane first touches the cone, at the vertex distance e ÷ sin β from the apex.

  3. Plot the cut curve

    For several distances s between that point and the base arc, set off the angle sin β × acos(e ÷ (s sin β)) inward from each seam line and mark the point. Join the marks with a smooth curve on both sides.

  4. Lay out the patch

    On a separate sheet, draw the base line 2√(r² − e²) long and the patch height (r − e) ÷ tan β. Plot its curved sides as half-width √((s sin β)² − e²) at height (L − s) cos β, using the same distances s.

  5. Check and allow

    The patch's curved edge and the cone's cut edge must be the same length. Add allowance on the edges you will join.

Good to know

  • FabLayout runs the same code as Trueflat; the result is exact geometry.
  • Source: Kittredge, The New Metal Worker Pattern Book (1917), Problem 147, Fig. 498, p. 278

Method and formula

The plane cuts a hyperbola out of the cone. What remains of the cone still unrolls exactly into a sector; the opening is closed by a flat patch whose outline is that same hyperbola.

  • Slant length L = √(r² + H²), sin β = r ÷ L, sector angle 360° × sin β.
  • Vertex of the cut at s = e ÷ sin β from the apex, where e is the plane's distance from the axis.
  • Cut edge at distance s from the apex lies sin β × acos(e ÷ (s sin β)) in from the seam line.
  • At the base the cut takes acos(e ÷ r) of the plan circle on each side.

Trueflat puts the cone's seam through the middle of the cut, so the cone is one piece for every cut distance. The plane may sit at most 0.45 D from the axis; beyond that the sliced-off cap is so thin that the cut curve and patch degenerate.

Worked example

D = 60 mm, H = 40 mm, e = 15 mm: L = 50 mm, sin β = 0.6, sector angle 216°. The cut starts 25 mm from the apex and reaches the base at 36° in from each seam line (60° of the plan circle). The patch is 51.96 mm wide at the base and 20 mm high. Areas: cone 3791.11 mm², patch 644.14 mm².

Questions

When do I need this shape?

When a conical hood, hopper or cap has to sit flat against a wall or a neighbouring part. The plane parallel to the axis is the wall; the patch is the flat side that seals against it.

What happens when the cut goes through the axis?

With e = 0 you get exactly half a cone. The cut edges become two straight lines from the apex, and the patch is a triangle with the base diameter as its base and the cone height as its height.

Why is the seam on the cut side?

The book puts the seam on the back element. Trueflat puts it through the middle of the cut, so the cone stays one piece for any cut distance. The surface is the same; only the cut line moves.

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