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### Cubic Footage of a Tapered Log

```Date: 08/07/2002 at 18:42:15
From: Christopher Watkins
Subject: Cubic Footage of a Tapered Log

Is there an equation that will calculate the cubic footage of a log
given the diameters of each end and the length?

Currently, I am using Smalian's Formula:
(Area1 + Area2)/2 x Length (ft)
Area = .005454 x diameter^2

Is there a formula that is more accurate by accounting for the taper
in the log?

Example:
Diameter1 = 9.2 inches
Diameter2 = 10.3 inches
Length    = 102.84 inches
Smalian's calculates 4.443 cubic feet.

```

```
Date: 08/07/2002 at 23:06:05
From: Doctor Peterson
Subject: Re: Cubic Footage of a Tapered Log

Hi, Christopher.

A "tapered cylinder" like this is properly called a "frustum of a
cone," and the formulas for it can be found in the Dr. Math Formulas
FAQ:

http://mathforum.org/dr.math/faq/formulas/faq.cone.html#conefrustum

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

V = 3.14(4.6^2 + 4.6*5.15 + 5.5^2)*102.84/3 = 7686 in^3

Dividing this by 12^3, we get 4.448 ft^3. So your approximation is
pretty accurate in this case - particularly considering that a log is
probably not exactly conical anyway.

Out of curiosity, I looked up "Smalian" to see where your formula
comes from. It's clear why it would be close, since it is simply the
average of the volumes of cylinders with the maximum and minimum
diameters. The following page

Week 2 Mensuration Notes - Hans Zuuring, School of Forestry,
Univ. of Montana

explains that Smalian's formula applies to a frustum of a paraboloid,
which is slightly convex compared to a cone. It also gives a formula
equivalent to ours for the frustum of a cone, and one ascribed to
Newton that applies more generally, by using a measurement at the
middle to account for the actual curvature of the taper. I haven't
looked into just when each of these would be exact. This page gives
some more detail you may find useful:

http://sres.anu.edu.au/associated/mensuration/BrackandWood1998/
Select LOGS.HTM (Measurement of logs), and then volume.

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

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 08/08/2002 at 11:33:12
From: Christopher Watkins
Subject: Thank you (Cubic Footage of a Tapered Log)

Dr. Peterson,
Thank you so much for your time, and for the quick lesson.
For your information, I work for a timber company, and a
computer scanner at our lathe (plywood) is calculating a
higher cubic ftg number than I was physically calculating.
I believe that this is due to an offset in the computer
system, but I wanted to make sure that I was using the most
accurate formula before I submitted my report.

I really appreciate your time, and apologize for not
finding the formula before writing.  I was not aware that
the situation or example is termed a "frustum".  I was
looking in other areas.
Wonderful Job!!
```

```
Date: 08/08/2002 at 12:09:59
From: Doctor Peterson
Subject: Re: Thank you (Cubic Footage of a Tapered Log)

Hi, Christopher.

I was intrigued by the formulas I found researching your question,
because I found information about formulas for the parabolic frustum
only in forestry sites, which didn't make it clear whether they are
exact or approximations (since they seem so different). Our site
doesn't even list the formula for volume of a segment of a paraboloid,

V = pi r^2 h/2

Looking a little deeper than I did last night, I found that
"Smalian's formula" is correct for a parabolic frustum

V = pi (r1^2 + r2^2) h/2
or
V = h(A1 + A2)/2

(where h is the height, r1 and r2 are the bottom and top radii, and
A1 and A2 are the bottom and top cross-section areas), while "Huber's
formula"

V = pi rm^2 h
or
V = h Am

(where rm and Am are the radius and area of the section halfway
between the top and bottom) is also correct, because for a paraboloid

rm^2 = (r1^2 + r2^2)/2

Further, "Newton's formula"

V = pi (r1^2 + 4rm^2 + r2^2) h/6
or
V = h(A1 + 4Am + A2)/6

is valid for both the parabolic frustum, the conical frustum, and in
fact any shape (called a general prismatoid) whose cross-sectional
area follows a linear, quadratic, or cubic law, which also includes
spherical segments and more:

http://mathworld.wolfram.com/GeneralPrismatoid.html

All this is beyond what you need, of course; I just wanted to fill in
some gaps in my list of formulas.

If you need any further help figuring out where the errors are
arising, feel free to send me additional information.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Geometry
College Higher-Dimensional Geometry
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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