Smalian formula – log volume from both end diameters

When you cannot reach the middle of the log – on a deck or a truck – you measure both ends.

What is the Smalian formula?

The Smalian formula calculates log volume as the average of the two end cross-section areas multiplied by the length. Both measurements are taken on the log ends, which makes it practical for decked or loaded logs. On logs with strong taper it slightly overestimates volume.

Formula

V = (A₁ + A₂) / 2 · L

A₁, A₂ = cross-section area at the small and large end (π/4 · d²) · L = length. In imperial units: V [ft³] = 0.002727 · (D₁² + D₂²) · L, diameters in inches, length in feet.

Worked example

A log with 26 cm and 34 cm end diameters and 4 m length: A₁ = 0.0531 m², A₂ = 0.0908 m², V = (0.0531 + 0.0908) / 2 · 4 = 0.288 m³.

Where it is used

USDA Forest Service cubic scaling, Canada (Alberta, B.C. Firmwood) and anywhere logs are measured on the ends of a deck.

Frequently asked questions

Is Smalian more accurate than Huber?

Not always. On typical logs Smalian slightly overestimates and Huber slightly underestimates volume. Smalian's advantage is that it does not need the mid diameter.

When should I use Smalian?

When logs are decked and the middle cannot be reached, or when the buyer requires it, as is common in North America.

How does it differ from Newton's formula?

Newton's formula also includes the mid cross-section weighted by 4: V = (A₁ + 4·Aₘ + A₂)/6 · L. It is the most accurate but needs three diameters. The app calculates it too.

Other log scaling methods

All 22 methods in the app ›

In your countryDoyle vs Scribner vs International ¼″Three US board foot rules, three different results for the same log. Clearly laid out, in numbers, with the reasons.BF · US · CA In your countryDoyle Log RuleThe oldest log rule still in wide use in the southern and midwestern United States.BF · US In your countryScribner Decimal CA rule built by drawing boards into circles, today the standard for softwood logs in the western United States.BF · US · CA In your countryInternational ¼″ Log RuleIt allows for a ¼-inch kerf, shrinkage and log taper, so it comes closest to the real lumber yield.BF · US Huber formulaThe most widely used scaling method in Central Europe: one diameter at mid-length and the length.m³ · SK · CZ · DE · AT · FR · IE JAS (Japanese standard)The small-end diameter squared method, the standard for logs traded to Japan.m³ · JP Swedish scaling (toppmätning)Swedish sawmills pay by the small-end diameter: m³to is a cylinder from the top end.m³ · SE Hoppus formulaA traditional British rule still used in the tropical hardwood and teak trade.m³ · GB · IE · IN · GH · MY More methodsSTN 48 0009, ČSN 48 0009, Vermont, Universal (Daniels), Newton, GOST 2708-75, Brereton, Ontario Board Foot, Ontario Cubic, Roy, Alberta, BC Firmwood, Mid-girth, Guo Biao (GB/T 4814)All 22 methods ›