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
   http://www.snr.missouri.edu/natr211/topics/volume.html 

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 
86.425.

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)
   http://www.utextension.utk.edu/pbfiles/PB1650.pdf 

   Volume and weight of harvested trees, and log rules (PDF file)
   http://forest.wisc.edu/facstaff/radeloff/No4_volume_logrules.pdf 

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
  http://mathforum.org/dr.math/