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Board Feet from a Log

Date: 03/19/2003 at 07:44:43
From: Jeff
Subject: Board feet from a log.

How do you compute board feet from a log?

What is the board feet of a 10-foot log if the diameter is 16 inches  
at one end and 14 inches at the other end?

I read in a book that the board feet can be computed by multiplying 
the length by the square of the diameter and then dividing by 24.

board feet = (ld^2)divided by 24

The book said to use the diameter of the small end of the log.

Using this I compute the board feet to be about 82.

However, a student of mine looked up the board feet on a chart off 
the internet, which said 63 board feet.

Please help.

Date: 03/19/2003 at 12:06:30
From: Doctor Peterson
Subject: Re: Board feet from a log.

Hi, Jeff.

The problem is that there are various ways to estimate the number of 
board feet in a log, none of which is really accurate. It is not a 
precise calculation of volume, since it takes into account losses due 
to kerf and making straight boards, and generally assumes rather than 
measuring the taper of a log. I did a Google search to find formulas 
for these rules, and the fullest discussion is at

   Natural Resource Biometrics 

This gives several rules, none of them as simple as what you used. 
The most accurate, the International Rule, with your example is

  0.44 D^2 - 1.20 D - 0.30 = 69.14 board feet

with D=14 inches for an 8 foot log. Since there is no rule given for 
a 10-foot log, we can scale this up by multiplying by 10/8 and get 

The widely used Doyle Rule is

  (D - 4)^2 * L/16 = 62.5

for D=14 inches and L=10 feet.

These two answers are similar to the two that you and your student 
got; yours is presumably more accurate.

Here are a couple other pages I found with other details:

   Understanding Log Scales and Log Rules (PDF file) 

   Volume and weight of harvested trees, and log rules (PDF file) 

The first of these compares the rules and shows how they are used. 
The second gives a brief summary and lists pros and cons, such as 
which rules tend to over- or underestimate.

Just to see how far off we would be if we calculated the actual volume 
of the log, let's use the formula for volume of a frustum:

  V = Pi(R^2 + rR + r^2)h/3

You have R=8 in, r=7 in, and L=10 ft. A board-foot is a foot^2-inch; 
so we can convert L to 120 inches and divide the volume we get by 144:

  V = pi(8^2 + 8*7 + 7^2)*120/3 = 21,237 in^3 = 147.5 board-feet

If I did that right, a lot is lost when you cut the log into boards! 
The extra 60 board feet is a lot of sawdust and scraps.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 

Date: 04/15/2009 at 12:50:08
From: David
Subject: Board Feet in a log discussion

Hi, I saw your dialog about board feet in a log and had a thought
or two coming at the discussion from the Logger/woodworker (I have 
some experience logging, managing wood lots and working in wood)
perspective in case you are interested.  

I would have guessed that the loss in milling the logs would be 40
or 45% but I think your calculations indicating a even higher percentage
could be close, too.  There are a few reasons that may be of interest. 

1) kerf size can be anywhere between 3/16 to 3/8 or more depending
on the type of blade.  

2) you cannot simply slice up a log and square off the edges to make
quality lumber although many mills that make poor lumber do.  For example,
you will never see the center rings of a log in the end grain of a #1
quality piece of lumber as such a board will warp excessively and move
a lot over time.  So, you should always reject such boards at Home Depot
or Lowes when buying larger dimensions of lumber.  

3) there are other demands from the market that create more waste such
as the width of most lumber is in even increments.  

4) most lumber is planed after it is milled creating more waste.  That
is one of the reasons that a 2x4 is really more like a 1 3/8 x 3 .

Hope all of this is helpful or at least interesting.  Peace to you.

Date: 04/15/2009 at 15:04:39
From: Doctor Peterson
Subject: Re: Board Feet in a log discussion

Hi, David.

Thanks for writing.

The waste I calculated in the example was something like 147.5 - 
86.4 = 61.1; as a percentage that is 61.1/147.5 = 41%, which fits 
right in your estimated ballpark. The lower estimate, 62.5, gives 
waste of about (147.5 - 62.5)/147.5 = 58%, which would be higher.

I always like to know the reasons for a formula, and your details 
help to show what is involved in what I called "a lot of sawdust and 
scraps"! In particular, this demonstrates the fact that, although 
math gives nice accurate numbers in theory, reality trims off a lot 
of the precision and makes rough estimates much more valuable than 
"exact" calculations.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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