Why dividing length by length gives the wrong answer
A cut list says you need twelve rails 18 in long. An 8 ft board is 96 in. Ninety-six divided by eighteen is 5.33, so five per board, so three boards. That answer is right by accident here and wrong most of the time, and it is wrong for two reasons that pull in the same direction.
The first is the kerf. Every crosscut turns a strip of wood into sawdust — 0.125 in for a full-kerf table saw blade, about 0.094 in for a thin-kerf, 0.04 in for a bandsaw. Cutting five 18 in parts takes four kerfs between them, so 90 in of parts actually eats 90.5 in of board.
The second is the end trim. Board ends check, split and get dinged in the rack, and a planer leaves snipe in the first and last few inches. Nobody uses the extreme ends of a rough board, so an inch or two comes off before the first useful cut.
Put both in and the arithmetic changes: 96 in of board minus 1 in of trim is 95 in of usable length, and 95 in holds five 18 in parts with 4.5 in left over. Still five — but move to 20 in parts and the same board holds four, not the 4.75 that plain division suggests, and you have thrown away 14.6 in per board.
The count is a floor function, so it changes in steps. That is why a quarter inch off a part length sometimes saves you an entire board and usually saves nothing at all.
The formula, and the +k that people leave out
Write L for the stock length, T for the end trim, p for the part length and k for the kerf. The usable length is L − T. Now ask what n parts actually consume: n lengths of part, plus the kerfs between them. There are n − 1 of those, not n, because the last part ends at the offcut and needs no cut after it.
So n parts fit when n·p + (n − 1)k ≤ L − T. Rearranging for n gives the form at the top of this page: n = ⌊(L − T + k) / (p + k)⌋. Writing it as ⌊(L − T) / (p + k)⌋ instead — which is the version most people reach for — undercounts by one part whenever the fit is tight, and one part per board across a big run is real money.
Ripping across the width works the same way, without the trim: rips = ⌊(W + k) / (w + k)⌋. A 5.5 in board gives two 2.5 in strips with a 0.125 in kerf, because 2 × 2.5 + 0.125 = 5.125 ≤ 5.5, while three would need 7.75 in. A 48 in plywood sheet gives four 11.25 in shelves, because 4 × 11.25 + 3 × 0.125 = 45.375 in fits and five would need 56.75 in.
Multiply the two counts for parts per board, divide your quantity by that and round up for the board count. Yield is finished part area over purchased board area — a blunt measure, but the one that tells you whether a different stock length would serve you better.
Everything here assumes identical parts. A real cut list with several part sizes is a bin-packing problem, and mixing sizes on one board almost always beats running each size separately.
Worked example: twelve 18 in rails from 1×6 pine
You need twelve rails, 18 in long and 2.5 in wide, from 1×6 pine at $24 for an 8 ft board. Your table saw has a full-kerf blade at 0.125 in and you trim 1 in off each board.
- Usable length. 96 − 1 = 95 in.
- Parts along the length. (95 + 0.125) ÷ (18 + 0.125) = 95.125 ÷ 18.125 = 5.248, floored to 5. Check it: five parts consume 5 × 18 + 4 × 0.125 = 90.5 in, which fits in 95; six would need 108.625 in, which does not.
- Offcut. 95 − 90.5 = 4.5 in of usable stock left on each board.
- Rips across the width. A 1×6 is 5.5 in wide. (5.5 + 0.125) ÷ (2.5 + 0.125) = 5.625 ÷ 2.625 = 2.143, floored to 2.
- Parts per board. 5 × 2 = 10.
- Boards required. ⌈12 ÷ 10⌉ = 2, and they yield 20 parts, so 8 are spare.
- Yield. Parts area is 12 × 18 × 2.5 = 540 in². Board area is 2 × 96 × 5.5 = 1,056 in². 540 ÷ 1,056 = 51.1%.
- Cost. 2 × $24 = $48.
Half the material becomes offcut, and the reason is visible in the numbers: the eight spare parts on the second board are 360 in² of that waste. Twelve parts is an awkward quantity for a board that yields ten. Twenty parts would come from the same two boards at 85.2% yield, and ten parts would come from one board at 85.2% as well — the calculation is 10 × 18 × 2.5 ÷ (96 × 5.5) = 450 ÷ 528.
Parts per board by part length, at 1/8 in kerf
| Part length (in) | Per 8 ft board | Offcut (in) | Per 12 ft board | Parts gained by going to 12 ft |
|---|---|---|---|---|
| 12 | 7 | 10.25 | 11 | 4 |
| 16 | 5 | 14.50 | 8 | 3 |
| 18 | 5 | 4.50 | 7 | 2 |
| 20 | 4 | 14.63 | 7 | 3 |
| 24 | 3 | 22.75 | 5 | 2 |
| 30 | 3 | 4.75 | 4 | 1 |
| 36 | 2 | 22.88 | 3 | 1 |
| 48 | 1 | 47.00 | 2 | 1 |
Read the offcut column before you buy. A 20 in part leaves 14.63 in stranded on every 8 ft board but only 2.25 in on a 12 ft board; a 30 in part leaves 4.75 in on the 8 ft and 22.63 in on the 12 ft. Which stock length wastes less depends entirely on your part length, not on the stock length alone.
How to read the yield figure
Yield tells you what fraction of the area you paid for ends up in the project. Above about 80% you are cutting efficiently and there is little left to win. Between 60% and 80% is normal for a small run of one part size. Below 50% something structural is wrong: either the part length strands a long offcut on every board, or your quantity is just over a multiple of the parts-per-board figure and the last board is mostly spare.
The stock-length table the calculator builds is the tool for the first problem. It runs the identical arithmetic against 6, 8, 10, 12 and 16 ft stock and shows what each yields. Do not read it as a price comparison unless you substitute the real price of each length, because it prices every row at the same figure you entered.
For the second problem — the last board being mostly spare — the fix is usually to find another part in the project that fits the offcut. If a board leaves 22 in after three 24 in parts, and your project also wants 10 in blocks, those blocks are free. That is the whole reason cabinet shops cut from a combined list rather than part by part.
Board footage is a different question from part count. If you are buying rough hardwood sold by the board foot rather than by the stick, price the job with the board feet calculator and use this page only for the count. And if the parts are shelves, check them for stiffness with the shelf sag calculator before you commit to a thickness — a thinner part that saves a board and then sags is not a saving.
Assumptions and limits
- All parts are identical. Mixing several part lengths on one board is a packing problem this page does not solve, and mixing almost always beats running each size separately.
- Rips and crosscuts are independent. The parts-per-board figure assumes you can rip the full length and then crosscut each strip. If you crosscut first, the count is the same but the offcut is distributed differently.
- Defects are not modelled. Knots, checks, wane and bow all remove usable length unpredictably. Rough hardwood commonly loses 10 to 20% beyond the arithmetic here, which is one reason lumberyards quote a separate cutting yield.
- Grain direction is ignored. On plywood the face grain runs the long way, so shelves ripped across the sheet are weaker than shelves cut along it even though the yield arithmetic is identical.
- The end trim is one number per board. If you square one end and leave the other, halve it. If both ends are checked, 2 in is more realistic than 1 in.
- Wood moves after cutting. Parts cut oversize now and milled to final size later need extra length and width; work out how much with the wood movement calculator.
Buy one more board than the arithmetic says
The count above is exact given perfect stock and a perfect operator. Neither exists. A tear-out on the last part, a board that turns out to be cupped, or a dimension you read wrong on the plan all cost you a part, and going back to the yard costs an hour. On any job where the parts are visible, buy a spare board and return it if you do not need it.
Where cut-list arithmetic sits among the alternatives
What this page solves is the one-dimensional cutting-stock problem restricted to a single part size — the simplest member of a family that gets hard quickly. Add a second part length and you are choosing which parts to pair on each board; add twenty and you are running an optimiser. Dedicated cut-list software does exactly that, and for a kitchen's worth of sheet goods it will beat hand planning by several sheets. For a project with one or two part sizes, the arithmetic here gives the same answer in a fraction of the time.
Sheet goods deserve a separate thought. Plywood, MDF and melamine come in fixed 48 × 96 in sheets, and the two-dimensional packing problem has a different character from ripping a board: you are choosing whether to cut the sheet the long way or the short way first, and that choice changes the yield without changing any dimension. The fabric cutting layout calculator solves the same geometry for cloth, where the width is fixed and the length is what you buy.
One more limitation worth naming: this page counts parts, not board feet, and lumber is priced both ways. Softwood dimensional stock is sold by the stick at a fixed price per length, which is what the cost figure here assumes. Rough hardwood is sold by the board foot at a price that varies with thickness and grade, and a hardwood dealer will often sell you a random-width, random-length pile from which the yield is genuinely unpredictable. In that world the count you compute here is a lower bound on what you should buy, not a shopping list.