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Truncated cone by triangulation calculator

The same reducer as the truncated cone page, laid out with triangles instead of two long arcs. Use it when the apex is too far away for your trammel.

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912.1 mm522.5 mm
Flat pattern: cut on the bold outline, bend on the dashed blue lines, the thin lines are marking lines

Outer pattern radius

522.8mm

Approximate, within ±3.6 mm

Inner pattern radius
In the app
Sector angle
In the app
Seam
In the app
Slant length
In the app
Cut 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 both ends

    Divide the large and the small circle into the same number of equal parts and number the points around each circle in the same direction.

  2. Find the true lengths

    Every straight element from a large-end point to the small-end point above it has the slant length. Find the diagonal from each large-end point to the next small-end point as the hypotenuse of a right triangle whose height is the cone height.

  3. Lay out the first triangle

    Draw one element at the slant length. From its small end swing the small-end chord, from its large end swing the diagonal; where the arcs cross is the next small-end point.

  4. Step around the strip

    From the large end swing the large-end chord and from the new small-end point swing the slant length to get the next large-end point. Repeat, alternating triangles, until every division is used.

  5. Draw the edges and close

    Bend a batten through the points on each edge, draw the last element as the second seam line and add your seam allowance.

Good to know

  • This pattern is approximate; the result card shows how close it is.
  • Source: Kidder, Triangulation Applied to Sheet Metal Pattern Cutting (1917), Chapter VI, "The Pattern for the Frustum of an Oblique Cone", Fig. 30, p. 42 (same triangulation, shown on an oblique cone). Cross-checked against the radial method of Daugherty, Sheet-Metal Pattern Drafting (1922), p. 66

Method and formula

Here the truncated cone is cut into a strip of triangles. Each end circle is divided into n equal parts, which turns the cone into an n-sided pyramid frustum, and every triangle is laid down on the sheet from its true side lengths. No centre point and no trammel are needed.

  • Element s = √((R − r)² + H²), the same for every division.
  • Chords 2R sin(180°/n) on the large end and 2r sin(180°/n) on the small end.
  • Diagonal √((R − r)² + 4Rr sin²(180°/n) + H²).

The radial method on the truncated cone page is exact. Triangulation is close but not exact: straight chords are a little shorter than the arcs, so each end of the strip comes out short by at most π³r ÷ (3n²), where r is that end's radius. The pattern radii and sector angle in the results are those of the true cone; the triangle strip closes slightly inside that angle. More divisions shrink the error by the square.

Worked example

Large diameter 60 mm, small 30 mm, height 20 mm, n = 6: s = 25 mm, chords 30 and 15 mm, diagonal √1075 = 32.79 mm. Six chords of 30 mm give 180 mm against a large-end circumference of 188.50 mm, so the large end is 8.50 mm short, inside the bound of 8.61 mm.

Questions

When should I use triangulation instead of the radial method?

Use the radial method when you can swing the pattern radius: it is exact. Use triangulation when the two diameters are close, so the apex is metres away, or when the pattern does not fit around one centre on your sheet. Triangulation only needs short measurements: one element, two chords and one diagonal.

How accurate is the triangulated pattern?

Each end comes out short by at most π³r ÷ (3n²), where r is that end's radius and n the number of divisions. Doubling n cuts the shortfall to a quarter. In the 60/30/20 mm example with n = 6 the large end is 8.50 mm short. Use more divisions when the fit matters, or the radial method when you can swing the radius.

Why do all the elements have the same length?

The cone is round and concentric, so every element from the large end to the small end is a slant height. Only an eccentric or oblique cone needs a separate true length for each element.

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