Calculate volume of wooden logs by top and bottom perimeter length and total length

Question

I bough wooden logs (not the chocks, sorry) and want to check the total volume of wood (m³).

I measured top perimeter length (circumference), bottom perimeter length and length of the even log.

The log section isn't always looks like a perfect circle, sometimes it's have an oval shape or a complex shape, but never curved inside.

So i have a three numbers in cm: 67, 68, 38. I entered all the data to spreadsheet and stuck with the right formula.

What the formula to find the volume of the even log?

Thank you.

Answer 1

What you want, it appears to me,is a portion of a toroid or a cylinder but not a chock.

By one theorem of Pappu, the swept out volume by rotation of areas around a central axis equals cross section area times central length $L$.From perimeters we compute radius at each end and swept volume.

$$ R=67/(2\pi) = \,10.663 {\,cm}, r = 68/ (2 \pi) = 10.823 {\,cm},\, L = 38 {\,cm},\, Vol= \pi ((R+r)/2)^2 \,L $$

So,Volume

$$= 13778\, {cm}^{3}$$

EDIT1:

WE can also compute the volume as a frustum of a cone

$$ \dfrac{\pi L}{3}(R^2+r^2+R r) $$

which also gives approximately the same answer.

Answer 2

I'm not sure how much inaccuracy you tolerate in your calculation, but you have to make some assumptions to simplify the situation. If you model your logs as stumpy cones, then use those formulas here (its german but you just need to look to the picture ;)): https://de.wikipedia.org/wiki/Kegelstumpf

If this model is oversimplified, then I see no way around to make a statistical analysis of your logs. You have to measure the volume (e.g. with a bucket of water) of many at most different logs and collect the volumes with the feature variables (height, radii, etc..) in table. Now you could use some classification algorithms (https://en.wikipedia.org/wiki/Category:Classification_algorithms). The simplest way to estimate the volume of a new (unknown) log is the nearest neighbour algorithm. For examples you had measured $n$ logs with the dimensions $(h_i,r_i,R_i)_{i=1\ldots n}$ and the corresponding volumes $(V_i)_{i=1\ldots n}$. You can estimate a volume of a new log $(h,r,R)$ by calculating $$i_{est}:=\operatorname{argmin}_{i=1\ldots n} d\left((h_i,r_i,R_i),(h,r,R)\right)$$ with your preferred metric $d$ (e.g. the euclidean norm $d(x,y):=\|x-y\|_2$). The estimated volume (regarding this algorithm) is given by $V_{i_{est}}$. The precision of this estimation depends crucially on the chosen metric, the amount of features you measure in your objects and the amount and choice of samples you fill into your table, so there is much space for experiments there. But if its done well, you will get a very reliable model!

EDIT: The second method is just recommendable if you plan to calculate volumes of many logs for accounting purposes or something like that.