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Elliptical cone flat pattern calculator

A cone or truncated cone on an elliptical base with the apex over its centre. Enter both axes, the height and the top-to-bottom ratio; you get the pattern around the apex.

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529.5 mm275.6 mm
Flat pattern: cut on the bold outline, bend on the dashed blue lines, the thin lines are marking lines

Seam

543.5mm

Approximate, within ±0.2 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 ellipse

    Divide the base ellipse into an even number of parts, using equal steps of the eccentric angle: x = a sin θ, y = b cos θ.

  2. Find the true lengths

    For each point, the element length to the apex is √(x² + y² + h²), where h is the full apex height.

  3. Lay out from the apex

    Draw the shortest element, at the end of the minor axis, from a centre point. Swing each next element length from the centre and the base chord from the last point to place the next point.

  4. Mark the top for a truncated cone

    If the top is cut off, mark the top end on each element at the top-to-bottom ratio times its length.

  5. Draw the edges and seam

    Bend a batten through the points, close along the shortest element 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 132, Figs. 473–475, p. 261. The book approximates the ellipse with circular arcs; Trueflat develops the true ellipse, so the method is related but not the same

Method and formula

The apex stands above the centre of the ellipse, so every element still runs to one point and the surface unrolls exactly. The elements differ in length: shortest at the ends of the minor axis, longest at the ends of the major axis.

  • Apex height h = H ÷ (1 − k), where k is the top-to-bottom ratio (0 for a full cone).
  • Element length √(a² sin² θ + b² cos² θ + h²), with a and b the half-axes.
  • Top edge at k times each element length; the top is an ellipse of the same shape.

The angle between elements on the pattern has no closed formula; Trueflat integrates it numerically, so the unrolled base edge equals the ellipse's perimeter. The axes may differ by up to 6 to 1. Thickness is taken to the middle of the sheet only.

Worked example

Major axis 80 mm, minor 40 mm, height 50 mm, full cone: shortest element √(20² + 50²) = 53.85 mm, longest √(40² + 50²) = 64.03 mm. The pattern spans 186.09°, and its base edge is 193.77 mm, the perimeter of the ellipse. With a top ratio of 0.5 the apex is 100 mm up and the pattern spans 105.45°.

Questions

How is this different from the book's oval method?

Pattern books draw the ellipse as an oval of four circular arcs and take the radii from a diagram. Trueflat uses the true ellipse, so the pattern fits a base that was cut as a real ellipse. If your base is a four-arc oval, the two will differ slightly.

Is the pattern exact?

The surface is a cone, so it unrolls without stretching. The angles have no closed formula; Trueflat integrates them numerically and draws the edges as straight chords through the curve samples. More samples bring the drawn edges closer to the true curves.

Where does the seam go?

On the shortest element, at one end of the minor axis. The pattern is symmetric about the opposite element.

What does the top-to-bottom ratio do?

At 0 you get a full cone with a point. Between 0.1 and 0.95 the cone is cut off at the given height, and the top opening is the base ellipse scaled by that ratio.

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