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Smalian's cubic log volume

Board-foot rules estimate how much lumber a log will make. Smalian's formula measures how much wood is in it. That distinction is the difference between an estimate somebody drew in 1846 and a piece of solid geometry - and it is why pulp, chip and export trade is done in cubic units.

All forestry calculators

Cubic volume from both ends

Measure both end diameters inside the bark and the length between them. Smalian's formula averages the two end areas and multiplies by the length - the standard for cubic scaling worldwide.

Units

Average the short and long axis at the butt end.

Smalian slightly overstates a strongly tapered log, because averaging the ends treats it as a truncated cone when the real stem profile is closer to a paraboloid. On straight sawlogs cut to length the error is small; on a whole tapered stem, scale it in segments.

Enter the large-end diameter, the small-end diameter and the length to get the cubic volume of the log.

The formulas on this page

Smalian's formula
V = ((A_b + A_t) / 2) x L
Mean of the two end areas, times the length. A_b is the large end, A_t the small end.
End area, imperial
A = 0.005454 x D²
Diameter in inches, area in square feet, volume in cubic feet.
End area, metric
A = 0.00007854 x D²
Diameter in centimetres, area in square metres, volume in cubic metres.

Measuring the log

  1. Both ends, inside bark

    Smalian's needs the large end and the small end. Measure inside the bark at both, and average the short and long axis on an end that is out of round.

  2. Length between the measured faces

    The length that goes in the formula is the distance between the two ends you measured - not the nominal scaling length with its trim allowance.

  3. Cut long stems into sections

    Smalian's is most accurate on short, straight, evenly tapered pieces. On a long or butt-swelled stem, measure it in sections and add the sections together.

  4. Keep your units together

    Inches with feet gives cubic feet; centimetres with metres gives cubic metres. The calculator will not let you mix them, and neither should a tally sheet.

A measure of wood, not of lumber

A cubic foot of wood is a cubic foot of wood. It does not depend on how somebody imagines the log being sawn, how wide the saw kerf is, or which century the rule was written in. That is Smalian's appeal: it is geometry, and two people measuring the same log will get the same answer.

It is also why cubic measure is what the pulp, chip, panel and export trades run on, and why research and inventory work is done in cubic units even in countries that sell lumber by the board foot. When the wood is going to be chipped, pulped or shipped, nobody cares how many boards were theoretically inside it.

What it will not do is tell you what a sawmill will recover. A 20-inch log and a 20-inch log with a hollow centre and a sweep in the middle can hold the same cubic volume and produce wildly different amounts of lumber. That is what the board-foot rules and a scaler's defect deductions are for.

The one thing to know about the formula

Averaging the two end areas treats the log as a truncated cone - a straight taper from one end to the other. Real stems are closer to a paraboloid, fuller in the middle than a straight line between the ends would suggest, which means Smalian's slightly overstates a strongly tapered piece.

On sawlogs cut to length from the middle of a straight stem the error is small enough to ignore. On a whole butt log with pronounced flare, or on a long tree-length stem, it is not. The fix is the one in the field guide above: measure in sections and add them, which is a better approximation of the real profile the shorter each section gets.

Huber's formula, which uses a single mid-length diameter, is the usual alternative and has the opposite bias. It needs you to reach the middle of the log to measure it, which is why Smalian's is what gets used in the woods.

Attributed to Smalian
The formula carries the name of a German forest mathematician and dates from the first half of the 19th century. This page attributes it rather than dating it precisely, because a primary source could not be checked.
Cubic feet and cubic metres
Both are real units of wood, so this calculator offers both - unlike the board-foot rules, which are defined on inches and feet alone.
Inside bark
Bark is not wood, and it varies enormously by species and age. Every measure on this page is inside the bark.
No defect deduction
This is gross volume. Rot, hollow, ring shake and sweep are deducted by a scaler looking at the log, not by a formula.

Cords, tons and why they are not the same conversation

A cord is a stacked volume - 128 cubic feet of space, including the air between the sticks - so the solid wood in a cord depends on how straight the pieces are and how well they were stacked. Published conversions to solid cubic feet exist and vary, which is exactly why the trade has moved toward weight. The firewood page works the cord through in full, since that is the market where a stacked volume is still what changes hands.

Weight is what most Southern pulpwood is actually sold on: the truck crosses a scale at the mill and the ticket is in tons. Tons do not convert reliably to cubic volume or to board feet, because green weight depends on species and on how long the wood has been on the ground drying. If your contract is in tons, the mill scale is the measurement that matters and nothing on this page replaces it.

Straight answers

Questions this calculator gets asked

Should I use this or a board-foot rule?

Use whichever your contract is written in. If you are selling sawlogs on a board-foot scale, the log rules calculator is the one that matches your contract. If you are selling pulp, chip or export wood, or you want to know how much wood is actually there, this one is.

Running both on the same log is a genuinely useful exercise, and it is how you learn what a board-foot rule is really doing to your small logs.

My log is bent. What do I do?

Measure it in sections along the curve rather than end to end, and add the sections. Sweep costs you volume in the mill even when the wood is there, so a scaler will also deduct for it - that deduction is a judgement about lumber recovery, not about cubic volume, and this formula does not attempt it.

Can I get board feet from a cubic volume?

Not reliably, and this site will not offer a conversion factor. The relationship between cubic volume and board-foot scale depends on which log rule you are converting to, on log size, and on taper - which is another way of saying the two measure different things.

If you need board feet, scale the log on the rule your contract names.

What these calculators are, and are not

  • They are arithmetic on numbers you type. Nothing is stored, nothing is sent anywhere, and nothing here has seen your property.
  • They are estimates, not an appraisal, a cruise or a scale ticket. When money changes hands, the measurement should come from somebody whose name is on it.
  • They are not investment, tax or legal advice, and nothing on these pages is an offer to buy or sell timber.
  • They carry no price, rate, cost or yield assumption of any kind. Every input is yours, which is why nothing here can go stale.

Where these formulas come from

Every formula on this page is published forestry mathematics, not an AXE USA invention. These are the extension-service and Forest Service publications that teach them, with the date each publisher shows. Retrieved and checked 14 August 2026.

  1. Purdue University Extension

    Log and Tree Scaling Techniques (FNR-191) (opens in a new tab)

    No publication date printed on the copy retrieved; the file's own metadata records it as prepared in December 2001

    Works through Doyle, Scribner and International side by side with worked examples, and explains why the three disagree. If you read one thing after using the log rules calculator, read this.

  2. USDA Forest Service Research and Development

    Forest Service research publications (opens in a new tab)

    Archive, continuously updated

    Where Reineke's 1933 stand density index and the density-management literature built on it live. Maximum stand density index by species and region comes from work published here and by the regional stations - never from a default typed into a website.

  3. Mississippi State University Extension Service

    Extension publications catalogue - forestry (opens in a new tab)

    Catalogue, continuously updated

    The source the owner's brief credits for the basal area formulation used on this site. MSU Extension's forestry publications are the plainest written introduction to stand measurement - and to log scaling - that a Southern landowner can get free of charge.

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