How to lay out a rectangular hopper by hand

What you are making

The worked example: a 600 × 500 bottom opening, a 200 × 150 top opening, 400 apart, first centred and then with the top shifted 100 to the right and 50 to the back. The openings you enter are the surface that gets developed; decide whether you are working to the inside, outside or mid-thickness and stay with it.

Why each panel is flat

Each panel joins one side of the bottom opening to the matching side of the top opening. Both sides run in the same direction, so they are parallel, and two parallel lines always lie in one plane: every panel is a flat trapezoid, for any offset. That is what makes the hand layout exact. (If the openings were not parallel, the plates would twist and be laid out by triangulation along a diagonal — Atkins, pp. 75–76; a top turned diagonally in plan is built from triangles too — Atkins, p. 77.)

1. True height of each panel

The panel's height on the flat sheet is not the hopper's height but the panel's slant: the hypotenuse of the vertical height and how far that side steps in, in plan — Atkins's “width of plate = √(height² + overhang²)” (p. 68). On the centred example the front side steps in (500 − 150) ÷ 2 = 175, so its true height is √(400² + 175²) = 436.61; the right side steps in (600 − 200) ÷ 2 = 200, so √(400² + 200²) = 447.21.

2. Lay out the panel

Draw the bottom edge, 600.00 for the front panel. Draw a line parallel to it at the true height. On a centred hopper the top edge sits in the middle of that line: its ends are 200.00 in from each end of the bottom edge, measured along it. Join the corners.

3. Check with the diagonal and the corner lines

Before cutting, measure across the panel from one bottom corner to the opposite top corner. It must equal that diagonal's true length in the part — the hypotenuse of the height and its plan length: for the front panel √(400² + 175² + 400²) = 592.14. The sloping sides must equal the corner lines, √(400² + 200² + 175²) = 480.23. If they do, the panel will meet its neighbours. This is also how the books build a panel: split by its diagonal into two triangles (Kittredge, p. 88; Kidder, pp. 5–6), each laid out from three true lengths, the diagonal's found from its plan length and the height (Neubecker, pp. 15–16).

4. When the top is offset

Shift the top 100 to the right and 50 to the back and the panels stop being symmetric. Each has its own true height (Atkins works an overhanging oblong hopper this way, Fig. 55, pp. 73–74), and its top edge is no longer centred on the bottom edge: it moves along the line by the plan offset in that direction. The front panel now steps in 225, so its true height is √(400² + 225²) = 458.94, and its top edge starts 300.00 from the bottom-left corner. The two sloping sides differ (548.29 and 469.71), and the diagonal check matters more than ever: 678.69.

“Top starts at” is measured along the bottom edge from its left-hand corner, looking at the panel from outside. A negative value means the top edge overhangs that corner.

5. How steep are the corners?

The panels of the centred hopper slope at 66.37° (front and back) and 63.43° (the ends) from horizontal, but the corner lines between them are flatter: a corner climbs the same 400 over both panels’ runs at once, √(200² + 175²) in plan, so it rises at only 56.40°. In general cot²C = cot²A + cot²B for the two panel slopes A and B — IS 9178 (Part 1) gives the same relation as the hopper’s valley angle (Fig. 5) and applies its slope rule for gravity flow to the valley, not just the walls, for square and rectangular hoppers (cl. 6.5); Maynard (p. 30) notes the valleys are the less steep part. At each corner the two panels meet at 100.33° on the inside, measured square to the corner line: the angle to set a fit-up jig or bevel gauge to. This is geometry only; it does not size a hopper for any material.

Common mistakes

• Using the hopper's vertical height as the panel height.
• Centring the top edge of an offset hopper's panels.
• Skipping the diagonal check — a panel can look right and still be a few millimetres out of square.
• Mixing inside and outside dimensions between the two openings.
• Checking a hopper’s slope on the panels only: the corners are never steeper, and flatter unless a panel is vertical.

How this matches the tool

The hopper tool develops each panel exactly from the same corner points, reports every edge, diagonal and true height, the slope of every corner line and the angle between panels, and exports a DXF per panel. Every number on this page was produced by its own code when the page was built.

Sources

1. E. A. Atkins, Practical Sheet and Plate Metal Work, 2nd ed. (Pitman, undated), Ch. IX, pp. 67–68: width of plate = √(height² + overhang²), equal tapering rectangular article
2. Atkins (2nd ed.), p. 70: oblong hopper of unequal overhang — sides and ends with their own slant heights
3. Atkins (2nd ed.), Fig. 55, pp. 73–74: overhanging oblong shoot or hopper
4. Atkins (2nd ed.), pp. 75–76: when top and bottom are not parallel, the plates twist and are laid out by triangulation along the diagonal
5. Atkins (2nd ed.), p. 77 “Twisted Square Base”: a square top twisted diagonally, built from triangles
6. G. W. Kittredge, The New Metal Worker Pattern Book (1901), p. 88: four-sided figures redivided by a diagonal into triangles, true lengths from diagrams of triangles
7. F. S. Kidder, Triangulation Applied to Sheet Metal Pattern Cutting (1917), pp. 5–6: a four-sided figure is fixed by its diagonal
8. W. Neubecker, Sheet-Metal Work (1920), pp. 15–16: true length of a panel diagonal from its plan length and the vertical height
9. IS 9178 (Part 1):1979, Criteria for Design of Steel Bins for Storage of Bulk Materials, cl. 2.15, cl. 6.5 and Fig. 5: the valley angle of a pyramidal hopper, cot²C = cot²A + cot²B
10. E. Maynard (Jenike & Johanson), “Ten Steps to an Effective Bin Design”, Chemical Engineering Progress, Nov. 2013, p. 30: valleys less steep than the side walls