What nesting looks like
One after another: the red triangles between the pieces are lost material
Nested: every second piece is flipped, and the angled cut is shared
The pieces are the same, but they take up less of the bar. The gain only becomes visible when the length left allows one more piece; the worked examples below show when that happens and when it does not.
How the angle and the cutting face are measured
The angle is the one between the cut and the length of the bar: 90° means a straight cut and 45° an angled one, as at the corners of a frame. The cutting face (F) is the side of the section on which the angle shows: on a 60×40 tube cut across the wide side it is 60 mm, across the narrow side 40 mm.
The gain at each shared joint is F divided by the tangent of the angle. At 45° it is exactly F (40 mm on a 40×40 tube); at 30° it is about 1.7 times F, and at 60° about 0.58 times F.
When two pieces can be nested
Two neighbouring pieces nest when the angles at the joint add up to 180°. A piece cut at 45° at both ends, flipped, has 135° at its ends, so it fits its neighbour. With straight cuts (90°) there is no gain.
Pieces with an angle at one end and its supplement at the other, for example 45° on the left and 135° on the right (a parallelogram), fit into each other even without being flipped.
A worked example
Values calculated with the formulas of the e-utilio Steel Cut Pro program, with 45° cuts at both ends, 6 m bars, a 3 mm blade and 40 pieces of the same length:
| Tube (face F) | Piece length | Pieces per bar, one after another | Pieces per bar, nested | Bars for 40 pieces, one after another | Bars for 40 pieces, nested |
|---|---|---|---|---|---|
| 40×40 (F = 40 mm) | 700 mm | 8 | 8 | 5 | 5 |
| 40×40 (F = 40 mm) | 900 mm | 6 | 6 | 7 | 7 |
| 40×40 (F = 40 mm) | 1,200 mm | 4 | 5 | 10 | 8 |
| 60×40 on the wide side (F = 60 mm) | 900 mm | 6 | 7 | 7 | 6 |
| 60×40 on the wide side (F = 60 mm) | 1,200 mm | 4 | 5 | 10 | 8 |
At 700 and 900 mm, with F = 40 mm, nothing changes. At 1,200 mm it goes from 4 to 5 pieces per bar, so 8 bars instead of 10. With a 60 mm face the gain is bigger: at 900 mm it goes from 6 to 7 pieces per bar and from 7 to 6 bars.
How to run the calculation on your own list
For a single piece you can use the quick calculator on the home page: enter the length, the cutting face and the angles and see how many pieces fit on a bar with and without nesting. For whole lists, with stock offcuts and several materials, use the cutting program; the user guide is here. More on reducing waste: cutting optimization.
Try the cutting program
Import your cut list and see the plan with nested pieces and the number of bars. Free demo, 3 days of full trial for new accounts.
Frequently asked questions
Does nesting work at any angle?
It works when the angles at the joint add up to 180°. With other combinations, the pieces do not have parallel cuts and cannot be fitted into each other.
Can I use angles greater than 90°?
Yes. An angle between 90° and 180° leans the cut the other way: for example, 135° at the right end matches 45° at the left end of the next piece.
What happens with the blade?
The blade is taken into account at every cut, so the plan does not exceed the length of the bar because of it.