Rectangular Transition development

What it is

The pyramid frustum: the reducer between two rectangular openings, on the same axis. It is the only transition in this catalogue that is not triangulated, and does not need to be — each of the four faces has its bottom edge parallel to its top one, so the face is a flat trapezoid and its pattern is the face itself, with no approximation.

Where it is used

Reducers in exhaust and air-conditioning ductwork, hoppers and silo outlets, feed chutes, transitions between rectangular-section equipment.

Measurements the calculation needs

• X Side X of the bottom mouth
• Y Side Y of the bottom mouth
• X2 Side X of the top mouth
• Y2 Side Y of the top mouth
• H Height between the mouths
• THK Plate thickness
• R Bend radius

How it develops: the formulas

Slant height of each pair of faces

hx = √( h² + ((Y − Y2)/2)² ) ; hy = √( h² + ((X − X2)/2)² ) hx is for the face with X at the bottom (it steps back (Y − Y2)/2) and hy for the face with Y. In the example (800 × 600 to 400 × 300, h = 500): hx = 522.0 mm and hy = 538.5 mm.

Each face

trapezoid of bases X and X2 (or Y and Y2) and height hx (or hy) All four faces are flat: the bottom edge is parallel to the top one. The template is exact, with no triangulation.

Angle between neighbouring faces

the dihedral comes from the 3D model In the example it is 96.1°: the angle measured with a protractor on the finished part, and the one the assembly note gives.

Worked example

The values the form comes pre-filled with, run through the tool itself:

Side X of the bottom mouth	800 mm
Side Y of the bottom mouth	600 mm
Side X of the top mouth	400 mm
Side Y of the top mouth	300 mm
Height between the mouths	500 mm
Plate thickness	3 mm
Bend radius	0 mm
The flat pattern fits a plate of	1681 × 1509 mm

Result of the example

Plate	1681 × 1509 mm

Assembly notes

The pattern develops the SHARP CORNER, on the mean line — the theoretical layout, which is what gets delivered. If you fold it on a press brake with an inside radius, the blank comes out LONG: each fold needs the deduction 2*(r+t)*tan(a/2) − a*(r + k*t), with r the inside radius, a the bend angle in radians, and your tooling's k. On a 500x300 tube in 1.5 mm sheet with four 90° folds that is 3.5 mm at radius 0.75 and 9.0 mm at radius 4.5 — always to be taken off. A welded corner from separate plates needs no deduction at all.

Fold to an angle of 96.1° BETWEEN THE TWO FACES — what a protractor reads on the finished part, not the press stroke angle. Every crease on this part has the same angle.

How to mark it out on the plate

1. Work out hx and hy. Mark the four faces in sequence, joined along the sloping edges, with the seam in the middle of a face or on an edge.
2. On each face mark the base (X or Y), go up hx or hy on the axis of symmetry and mark the smaller opening (X2 or Y2) centred.
3. Crease along the edges and fold to the angle between faces, checked with a protractor; the part closes with one seam.

Common mistakes and tolerances

Using one slant height on all four faces

They only coincide when the reduction is equal in X and Y. In the example hx and hy differ by 16.5 mm.

Folding with a radius and not deducting

The template is the sharp corner on the mean surface. Each fold with an inside radius needs the deduction 2 × (r + t) × tan(a/2) − a × (r + k × t); the assembly note gives the figure when r is entered.

Frequently asked questions

Why is this part not triangulated like the rectangle-to-round?

Because all four faces are flat. Two parallel lines always lie in one plane, and here the bottom edge of each face is parallel to its top edge — so there is no curved surface to approximate with triangles. On the rectangle-to-round one opening is a circle, and then no face is flat at all.

Is the slant height the same on all four faces?

No, and that is exactly where marking out from memory goes wrong. The face with X at the bottom steps back (Y − Y2)/2 and the face with Y steps back (X − X2)/2, so each pair has its own slant height. They only coincide when the reduction is equal in both directions.

Can the top opening be larger than the bottom one?

Yes, and in one direction only if you like — widening in X while narrowing in Y is routine in ductwork. The calculation does not distinguish a reducer from an expander.

What if both openings are the same?

You get a straight rectangular tube, and the pattern is a plain rectangle of perimeter by height. That is a real part, not a degenerate case, and the calculation accepts it.