FabLayout

Rounded corner hopper calculator

For a hopper or transition between two rectangles with rounded corners: enter both sizes, both corner radii and the height. One-piece pattern of flat sides and cone corners.

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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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1,654.0 mm855.2 mm
Flat pattern: cut on the bold outline, bend on the dashed blue lines, the thin lines are marking lines

Seam

1,077.0mm

Approximate, within ±1.9 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. Divide the corner arcs

    Divide each corner arc of the bottom and of the top into the same number of sections at the same angles.

  2. Join matching points

    Join each bottom point to the top point at the same angle; on every corner these lines meet at one point, the apex of a cone.

  3. Find the true lengths

    Work out the true length of each joining line and one diagonal per corner piece from the plan distance and H.

  4. Lay out from the seam

    Start with the seam in the middle of an A side, then add flat sides and corner pieces in turn.

  5. Close

    Finish on the second seam line, curve both edges through the corner points and add your allowances.

Good to know

  • This pattern is approximate; the result card shows how close it is.
  • Source: No published source checked for this shape. Method: flat trapezoid sides and truncated cone corners between the two openings, points joined at equal tangent angles, triangulated; checked against a separate reference implementation in Trueflat.

Method and formula

Each opening is a rectangle with rounded corners, the top centred over the bottom. The sides are flat trapezoids. Each corner is an oblique truncated cone between the two arcs, because arcs joined at equal tangent angles in parallel planes always meet at one apex. With equal radii a corner becomes an oblique cylinder; when the corner centres sit one above the other it is a right truncated cone.

  • Opening perimeter = 2(A + B) − (8 − 2π) r.
  • Radii: 0, or from 5 mm up to half the shorter side. At least one radius must be above 0; both 0 is a rectangle to rectangle transition.
  • Shallow, wide hoppers are allowed; the only slope limit is that no line runs more than 10 × H horizontally.
  • Accuracy: the corner arcs are drawn as n chords, so each edge is short by at most π³ × max(r1, r2) ÷ (48 n²).

With sheet thickness and inside or outside dimensions, each opening is moved by one amount to the middle of the sheet. This is approximate: the calculator shows how far the opening can be off, and asks for middle-of-sheet dimensions if that is more than 1 mm.

Worked example

Bottom 300 × 300 mm with 50 mm corners, top 500 × 500 mm with 150 mm corners, H = 300 mm, one section per corner. The corner centres coincide in plan at (100, 100), so each corner is a right truncated cone. The four sides are 4 × 200 × √(100² + 300²) = 252 982 mm² and the corners 174 356 mm², 427 338 mm² in all. Both seam edges total 632.46 mm; the exact perimeters are 1114.16 and 1742.48 mm.

Questions

How is it different from a rectangle to rectangle transition?

There the corners are sharp bends between flat sides. Here the corners are rolled cone sections between two radiused openings, with no sharp bend at the corners.

Can one end have sharp corners?

Yes. Set one radius to 0; that end's corners become cone apexes, as in a square to round. With a 0 mm bottom radius and a square top whose radius is half its side, the top is a circle and you get a square to round.

Is the pattern exact?

The flat sides and cone corners unroll exactly, but the corner arcs are drawn with chords. Each edge is short by at most π³ × max(r1, r2) ÷ (48 n²). Inside or outside dimensions with thickness are converted approximately and the deviation is shown.

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