What a board foot is, and what a log rule is trying to do
A board foot is a unit of lumber volume: 144 cubic inches, which is a piece one inch thick, twelve inches wide and one foot long. For sawn lumber it is exact arithmetic — thickness in inches times width in inches times length in feet, divided by twelve. A 2×8 twelve feet long is 2 × 8 × 12 ÷ 12 = 16 board feet.
A log rule is a different animal. It predicts how many board feet of lumber a sawyer will actually recover from a round log, which means it has to account for the slabs cut off the outside, the sawdust lost in each kerf, and the taper that makes the log's small end its limiting dimension. None of that is exact, so the answer depends on assumptions about saw thickness, board thickness and taper — and different rules make different assumptions. That is why three rules that all claim to measure the same log can differ by a factor of two.
Because the rule changes the number, it changes the money. A stand of small-diameter timber sold on Doyle scale and a stand sold on International scale are not being priced on the same basis, and a per-thousand price only means something once you know which rule produced the thousand. Sale contracts name the rule for exactly this reason, and so does every published price report.
The three rules and the assumptions inside them
Doyle is BF = ((D − 4) ÷ 4)² × L. The subtraction of four inches is a slab allowance and the division by four covers kerf and shrinkage. Its defining problem is that the four-inch allowance is a constant: it removes half the diameter of an eight-inch log and only a sixth of a twenty-four-inch one. Doyle therefore understates small logs severely and its shortfall shrinks as logs get bigger. Notice that at a 16-foot length the formula collapses to BF = (D − 4)², since the 16 in the numerator cancels the 16 in the denominator — a handy piece of mental arithmetic in the woods.
Scribner is a diagram rule: its author literally drew boards inside circles of each diameter and counted them, assuming one-inch boards and a quarter-inch kerf. Because it comes from drawings rather than an equation, its authoritative form is a table, and Scribner Decimal C — the version most widely used, which rounds volumes to the nearest ten board feet — is what mills scale on. The closed form used here, (0.79D² − 2D − 4) × L/16, is the published algebraic approximation to that diagram, and it can differ from the official table by several board feet, most noticeably on small logs.
The International ¼-inch rule is the only one of the three derived from a real accounting of saw geometry. It works in four-foot sections, allows half an inch of taper per section, computes 0.22d² − 0.71d for each section from its own small-end diameter, and multiplies the sum by 0.905 to convert the underlying eighth-inch-kerf rule to a quarter-inch kerf. Because it models the log rather than approximating it, forestry research and government inventories generally use International as the reference scale, and it is the closest of the three to actual mill recovery.
All three take the small-end diameter inside bark, because the small end limits what a straight board can be cut from, and bark is not lumber.
Worked example: ten 16-inch, 16-foot logs
You have ten sawlogs, each 16 feet long and 16 inches across the small end inside bark, and a buyer offering $450 per thousand board feet, Doyle scale.
- Doyle. (16 − 4) ÷ 4 = 3. Then 3² × 16 = 144 board feet per log. Since the length is 16 feet, the shortcut (16 − 4)² = 144 gives the same answer.
- Scribner. 0.79 × 16² = 0.79 × 256 = 202.24. Subtract 2 × 16 = 32 and then 4: 202.24 − 36 = 166.24 board feet. The length factor is 16 ÷ 16 = 1, so no further scaling.
- International ¼-inch. Four sections at 16, 16.5, 17 and 17.5 inches. Evaluating 0.22d² − 0.71d gives 44.96, 48.18, 51.51 and 54.95, which sum to 199.60. Multiply by 0.905: 180.64 board feet.
- The load on Doyle. 144 × 10 = 1,440 board feet, which is 1.440 MBF.
- Value. 1.440 × $450 = $648.00.
Now look at what the rule choice is worth. The same ten logs scale 1,662 board feet on Scribner and 1,806 on International. If the buyer's $450 offer were quoted on International scale instead, the load would fetch 1.806 × $450 = $812.88 — $164.88 more for exactly the same wood. Comparing offers means converting them onto one rule first: 144 ÷ 180.64 = 79.7%, so a Doyle price has to be about 1.255 times an International price to be equivalent on these logs.
How to read the three numbers against each other
The relationship between the rules is not a constant, and this is the part people get wrong. Doyle's shortfall against International is severe on small logs and narrows steadily as diameter grows. On a 16-foot log, Doyle gives 16 board feet where International gives 39 — 41% — at eight inches; 144 against 181, or 80%, at sixteen inches; and 400 against 424, or 94%, at twenty-four inches. Extend the comparison and Doyle actually crosses above International at about thirty inches of small-end diameter, where the two give 676 and 674 board feet respectively. So "Doyle underscales" is only true over the diameter range most sawlogs occupy, not universally.
This has a direct commercial consequence. If you are selling small-diameter timber, the rule named in the contract matters enormously and Doyle is the least favourable of the three to you. If you are selling large, clean butt logs, the three rules converge and the choice matters far less. Buyers know this, which is why regional custom about which rule is used tends to align with the timber type in that region.
Two more things to keep straight. First, everything on this page is gross scale. Mill scale deducts for rot, sweep, crook, shake and other defect, and a log with a hollow butt or a bad bend can lose a large share of its gross figure. Second, none of these rules is a measurement of wood volume. For cubic volume — used in pulpwood, in most of the world outside North America, and in growth-and-yield work — you use Smalian's or Huber's formula on the log's end areas instead, and the answer is in cubic feet or cubic metres with no recovery assumption in it at all.
The three rules compared on 16-foot logs
| Small-end diameter | Doyle | Scribner | International ¼-in | Doyle as % of Intl |
|---|---|---|---|---|
| 8 in | 16 | 31 | 39 | 41% |
| 10 in | 36 | 55 | 65 | 56% |
| 12 in | 64 | 86 | 97 | 66% |
| 14 in | 100 | 123 | 136 | 74% |
| 16 in | 144 | 166 | 181 | 80% |
| 18 in | 196 | 216 | 232 | 84% |
| 20 in | 256 | 272 | 290 | 88% |
| 24 in | 400 | 403 | 424 | 94% |
The last column is Doyle ÷ International, computed from the two columns beside it. It rises across the whole table, and continuing the same arithmetic to 30 inches gives Doyle 676 against International 674, where the two rules cross.
Mistakes that cost money in the woods
- Comparing prices across rules. A dollar figure per thousand board feet is meaningless until the rule is named. Convert both offers to one rule before deciding.
- Measuring outside the bark. Every rule expects diameter inside bark at the small end. On thick-barked species the difference is easily an inch or two, which on a small log is a large share of the volume.
- Measuring the large end. The small end limits what can be sawn straight through the log. Scaling on the butt overstates the volume badly on a tapered log.
- Forgetting trim allowance. Logs are bucked a few inches over the nominal length so the mill can square the ends. Scale the nominal length, not the tape length.
- Quoting gross scale as if it were net. Defect deductions come off after the rule, and on a defective log they can be a large fraction.
- Treating a formula as the official Scribner table. Scribner is a diagram rule and its authoritative form is a table, typically Decimal C rounded to ten board feet. The algebraic form here approximates it.
Where log rules sit among forest measurements
Log scaling is one step in a chain. Standing timber is cruised by measuring diameter at breast height and merchantable height, then converting through volume tables or equations; the tree height calculator covers the height side of that measurement and counts the 16-foot logs a stem will yield. Once a tree is felled and bucked, log rules take over. Once the lumber is sawn, board feet become exact arithmetic again.
Outside sawtimber, other units apply. Pulpwood and firewood are traded by the cord, a stacked volume of 128 cubic feet; the firewood cord calculator handles that conversion and the difference between a full cord and a face cord. Internationally, and increasingly in research within North America, roundwood is measured in cubic metres with no recovery assumption at all, which sidesteps the whole log-rule question.
If you are selling timber rather than buying it, three practical steps are worth more than any calculator. Get the rule and the scaling point written into the contract. Have the load scaled by someone independent, or at minimum tally and scale your own logs before they leave. And run a size-class inventory rather than an average, because these rules are strongly non-linear in diameter and the average log's scale is not the scale of the average.