Geometric vs. Arithmetic mean to determine diameter of object

I want to estimate the volume of irregular objects (coral fragments) by simplifying their structure to a cylinder and then calculate the volume of this cylinder (i.e. Ecological Volume).

The length of the cylinder runs along the longest part of the coral. Perpendicular to this I have measured the widest part (width1). Perpendicular to width1, I have measured a second width (width2). I want to use width1 and width2 to estimate the average diameter $d$ of the cylinder. Now this can be done in two ways:

1. $d = \frac{\text{width1} + \text{width2}}{2}$
2. $d = \sqrt{\text{width1} \times \text{width2}}$

I only recently realised that there are actually different types of means. For now, I am completely clueless as to which mean to use. Any input greatly appreciated!

For an example of a typical coral being measured, see: https://www.flickr.com/photos/ewoutknoester/29756453161/in/dateposted-public/

• Fractals are irregular, crinkly things that defy conventional measure, e.g., coastlines. Coral volume is also an example of a fractal. Given that a different, possibly more accurate approach would be to estimate the fractal dimension of these objects. There is a large literature out there on this. Good introductions include Gleick's book, Chaos, Mandelbrot's dense, Fractal Geometry of Nature, as well as the papers of James Theiler at MIT. – Mike Hunter Sep 29 '16 at 13:33
• The second approach (geometric average) is likely more appropriate, as noted by mdewey. What is the goal of the analysis? @DJohnson depending on the size of the fragments, and the goal of the analysis, the fractal component could be more or less significant. If a cylinder approximation is being used, likely the fragments are small and "arm-like", for example, so the volume approximation may not be bad. However the surface area associated with that volume will likely be much larger than for the cylinder, due to the fractal. (So e.g. mass approx. OK, but flux approx. way off) – GeoMatt22 Sep 29 '16 at 14:43
• @GeoMatt22: The aim of the experiment is to track the growth of corals in time, under different treatments. – EG Knoester Sep 29 '16 at 18:46
• @DJohnson: thanks for the new information! Alas, I already performed the measurements, so I'll have to deal with the data I have. – EG Knoester Sep 29 '16 at 18:46
• For the experiments you describe, it seems your methodology may be relatively imprecise. Do you have an idea of the effect-size expected under different treatments? (If you had photos like you posted, from a couple of angles, you could probably do much better with some image processing. Would have a learning curve though unless you have colleagues who already set up a system & workflow) – GeoMatt22 Sep 29 '16 at 19:15

You actually need the cross-sectional area to find the volume so you need to know $r^2$. If the cross-section was elliptical you would find its area by the formula $\pi a b$ where $a$ and $b$ are the semi-major and semi-minor axis lengths. So i would deduce that your second formula is closer to the truth since $a b$ fills the same role as $r^2$. Of course if coral is not approximately elliptical in cross-section this is only going to be very approximate.