Simple Explanation of Volume Formulas  

TOPIC: A simple explanation of the volume formulas

== 1 of 5 ==
Date: Sun, Dec 30 2007 4:49 pm
From: "Edward Frank"


There has been a recent flurry of posts concerning the geometry and calculus of various tree section forms. For those of you not into math I want to try to explain in more basic terms what these posts were talking about. There are four basic shapes that were being discussed that can be used to model portions of the trunks of trees for volume calculations. These are: Cylinder, paraboloid, cone, and neiloid. A cylinder is the most basic form. It looks like a can of soup in shape. The volume of a cylinder is the area of a flat end of the can time the height of the can. The paraboloid might be though of as the pointiest end of an egg in shape, with the tip of the point equating to the top of the tree. The next shape is a simple cone shape, with the pont at the top of the tree, The final form is the neiloid. The base of a tree often flairs outward in a concave shape. Think of the pointed sections in an egg carton that fit between the eggs. These are a neiloid in shape. In a tree these shapes are very tall and narrow, but are still basically the same as the examples.

How to relate these forms to volume is pretty simple if looking at the entire tree as a single shape. As mentioned above the volume of a cylinder is the area of the end of the can times the height of the can. Since the can, or tree is basically round in shape, then the area of the "end" of the can or a cross-section of the tree is (pi)r^2 (pie x radius squared). This is multiplied by the height. The other forms follow this same formula. The cross section of each is (pi)r^2. This in each case is multiplied by the height. The only basic difference is what is in the bottom of the formula. For a cylinder it is (pi)r^2 x height/1, for a paraboloid it is (pi)r^2 x height/2, for a cone it is (pi)r^2 x height/3, for a neiloid it is (pi)r^2 x height/4. The only difference is by what number you are dividing this basic formula. 1 - 2 -3- or 4.

Now what Bob has done with those unfriendly looking equations if to figure out how to apply these basic formulas to successive section of the tree as you move upward. Each measured girth becomes the base of a new shape. Therefore he can calculate which form best approximates that particular section of the tree. For the most part, the typical shape of most trunk sections is somewhere between that of a paraboloid and that of a cone - in effect the denominator of the equation is somewhere between 2 and 3 (if it was in a basic form.).

The final series of equations showed how being out of round would effect the volume formulas. Any tree that is not perfectly round will have a cross-section area that is somewhat less than a circle with the same girth. Therefore if you just use the diameter as calculated by dividing the girth by pi, you will overstate the volume of the tree to some degree depending on how out of round the tree is. Bob was showing how this difference varied with how out of round the tree was so that better volume calculations could be made.

Ed Frank
(I am trying to remember to delete old posts from my replies and be conscious of changes of topics)

== 3 of 5 ==
Date: Sun, Dec 30 2007 6:21 pm
From: "Edward Frank"


As I look over the post [above] I find I have made one of those hideously annoying "intuitively obvious" jumps that I always hated in math class. In the next to last paragraph I say: "For the most part, the typical shape of most trunk sections is somewhere between that of a paraboloid and that of a cone - in effect the denominator of the equation is somewhere between 2 and 3 (if it was in a basic form.). " This would be if that form were applied to the entire rest of the tree. For a particular segment of a cone or paraboloid, obviously the bottom part of a cone or paraboloid occupies more of the space than does the top part of the form. These numbers are for the average of the complete form from top to bottom in the cylinder. So, in the case of these segments, the shorter the segment the higher percentage of the volume of the general cylindrical shape they occupy, and the closer in size are the volumes for the cone and the paraboloid form to each other. [limit as height approaches 0 is 100%] Bob has made a series of tables/formulas that show what the upper diameter and the section volume would be for the segment when mixing different proportions of the cylindrical shape with that of the paraboloid shape, to obtain the best fit for the section.

The between 2 and 3 denominator reference is still applicable to the entire tree when the final volume is calculated. It is in fact the reciprocal (1/x) of the percentage that the actual volume of the trunk divided by the volume of a cylinder of equal basal diameter and height. This cylinder occupation concept has been mentioned briefly before, and I will do a more detailed write-up about it soon. The caveat here is if the tree has a particularly wide flaring base the percentage occupation will be low, and if the tree has a broken top, the cylinder occupation percentage will be higher.


== 4 of 5 ==
Date: Sun, Dec 30 2007 9:47 pm


Some added information to go along with Ed's lucid explanation is in order. Each geometric solid that we use in ENTS has three types of equations associated with it:

(1) An equation that expresses the total volume of the solid, e.g. V = (1/3)*pi*r^2 for the cone.

(2) One or more equations that express the volume of a segment of the solid formed between parallel planes passing through the solid at right angles to the axis of the solid. The segment is called a frustum.

(3) An equation that expresses the taper of the solid. From the taper equation, equations of classes (1) and (2) can be derived via integral calculus.

I successfully derived a general equation of taper that can be used to derive (1) and (2). The equation is:

r = R*[(H-h)/H]^p

Class (2) equations can take any of several forms. Each equation is based on one of two basic assumptions:

(a). The frustum is part of a single solid that covers the entire trunk,

(b). The frustum does not assume a single solid.

Much of the article that will be published in the spring edition of the ENTS Bulletin will present class (2) equations for the cone, paraboloid, and neiloid.


== 5 of 5 ==
Date: Mon, Dec 31 2007 1:03 am
From: Beth Koebel

Bob and Ed,

Thank you for a simple explanation of the math that
you have been talking about. I'm sure that I was not
the only one on the list that was getting totaly lost.
The most math that I had was simple Trig and a little