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Twisted rectangular transition calculator

For a rectangular transition whose top is turned relative to the bottom: enter both rectangles, the height and the twist angle. Each side folds along a diagonal into two flat triangles.

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Enter your measurements

Your dimensions are measured to the

Export from the Trueflat app

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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658.5 mm1,158.3 mm
Flat pattern: cut on the bold outline, bend on the dashed blue lines, the thin lines are marking lines

Seam

643.6mm

Approximate, within ±0.0 mm

Cut length
In the app
Top edge (chords)
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. Draw the plan

    Draw the bottom rectangle and, on the same centre, the top rectangle turned by the twist angle.

  2. Work out the true lengths

    For each side, find the true lengths of the corner line and of the diagonal from the next bottom corner to the top corner: √(plan distance² + H²).

  3. Build the triangles

    From the seam corner, lay out each side as two triangles: bottom edge, corner line and diagonal, then diagonal, top edge and the next corner line.

  4. Mark the folds and close

    Mark the corner lines and the diagonals as bend lines, finish on the seam corner line and add your seam allowance.

Good to know

  • This pattern is approximate; the result card shows how close it is.
  • Source: No published source checked for this shape. Method: each side is split along its outward diagonal into two plane triangles, and every triangle is laid out from three true lengths; checked against a separate reference implementation in Trueflat.

Method and formula

When the top is turned by an angle τ, the top and bottom edges of a side are no longer parallel, so a side cannot be one flat panel. Trueflat folds each side along the diagonal from the next bottom corner to the top corner, which gives eight flat triangles. This diagonal is an edge of the outer envelope of the two openings, so every fold points outward and the part has no dents.

The pattern of this folded part is exact. The other way to make the part, a smooth warped side hammered or stretched into shape, cannot be unrolled exactly. The deeper the diagonal fold, the further the part is from that warped shape.

  • Twist from 1° to 90°. For more than 90°, make two pieces in a row.
  • With long, narrow openings and a large twist (about 60° and up) the pattern can overlap itself at the fixed seam; the calculator then asks for less twist or different proportions.

With sheet thickness, inside or outside dimensions are converted to the middle of the sheet approximately, and the page shows by how much each opening can be off.

Worked example

Bottom and top both 400 × 400 mm, H = 300 mm, τ = 90°. Each top corner sits straight above the next bottom corner, so the pattern is eight right triangles with legs of 400 and 300 mm. Each seam edge is 500 mm, the area is 8 × 60 000 = 480 000 mm² and the cut length is 3200 mm.

Questions

Why is there a diagonal bend on every side?

With the top turned, a side's top and bottom edges are skew, so the side cannot be flat. Bending it along the diagonal makes two flat triangles that unroll exactly, and the bend always points outward.

Is the pattern exact?

For the folded part, yes: every triangle comes from three true lengths. A smooth warped side would have to be stretched and has no exact flat pattern. With thickness, inside or outside dimensions are converted approximately and the deviation is shown.

When should I use the rectangle to rectangle calculator instead?

When the top is not turned. A twist of 0° is a plain rectangle to rectangle transition with four flat trapezoid sides and no diagonal bends; this page starts at 1°.

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