## Estimating How Much Your Fish Weigh

Why would you want to know how much your fish weigh? I can think of three possible reasons: curiosity, estimating how much to feed them (based on the 2% of body weight rule), or for estimating a dosage of medicine. I'll offer two suggestions here that hinge upon figuring out the volume of your fish.

The reason for estimating volume is that goldfish typically, through the use of their swim bladder, maintain themselves at approximately neutral buoyancy. This means they neither sink nor float. It also means that we know, thanks to the insight of Archimedes, that they are displacing a volume of water equal to their mass. Since the density of water is well known (1 gram per milliliter is a good figure to use), an estimate of a fish's volume in milliliters will directly give its weight in grams.

So how do we measure the volume of a fish? Either directly or indirectly. To measure volume directly, get a measuring container (like a beaker), put in a measured amount of water, then put in a fish and note the change in volume. This gives you the volume of the fish. Then using the figure of 1 gram per milliliter you can convert the volume to a mass (or 1 ounce mass per fluid ounce, for those of you who prefer standard units). I would expect this method to yield a very accurate estimate.

To measure the volume indirectly, you can approximate your fish as an oblate ellipsoid, using its length width and height. An oblate ellipsoid is an ellipsoid where none of the three axes are of equal length. The equation for the volume of such a geometric solid is pi times length times height times width divided by 6 (or more simply 0.52xLxWxH). I would suggest using the snout-to-vent distance as the length.

Here are some estimated masses for given lengths for Ranchu-ish dimensioned fish (specifically the ratio of length to height to width is 1 to 0.7 to 0.5). Length is snout to vent and all measurements are for the body: no fins. The weight numbers were calculated using the indirect method.

 LENGTH (in.) MASS of FISH (oz.) 1 0.107 1.5 0.362 2 0.858 2.5 1.676 3 2.896 3.5 4.598 4 6.864 5 13.41

To validate the two methods against each other, I measured both the displacement and the dimensions of four Ranchu. To take the data, I prefilled a plastic "pasta storage" container (a tall and narrow shape) with tank water to a reference line. I then caught each fish, measured it, and placed it in the plastic container. I marked the new water level with the fish in the container and then dumped the fish back into the tank. To find the volume, I took a measuring cup marked in milliliters and used this to add a measured amount of water to the plastic container to fill from the reference line to the "with fish" line. That amount of water was the volume of the fish. The figures for the Ranchu measurements are in the table below. As a confused product of the American educational system, I freely mix metric and standard units. The table below is in centimeters and grams, for lengths and masses, respectively. The calculated mass (from the indirect method formula above) is the "Calc. Mass" column and the measured mass (from the displacement method) is the "Meas. Mass" column. The error is the percentage difference between these two columns.

 Fish Length Width Height Calc. Mass Meas. Mass Error Dana 11 5 6.5 187.2 208 10% Charlie 10 5 7 183.3 177 -4% Bravo 10.5 5.5 7.5 226.8 208 -9% Mikey 10.5 5 7 192.4 188 -2%

I was quite surprised at the agreement of the estimates. I would have to say that my dimensional measurements are probably accurate to 5% and my volume estimations to 5-10%. So to get 10% or closer agreement was much better than I had anticipated. I was also interested in how similar the sizes were for these 3-year old Ranchu. They are male siblings from the same spawn, but when I got them at 9 months of age, there were significant differences in their sizes. But over time they have converged to very similar dimensions.

The beauty of the indirect method for estimating goldfish weight is that once you have the approximate ratio of the dimensions for a particular variety (say Ranchu or Shubunkins), the only measurement you should need to get a reasonably accurate weight estimate is the length. Which means you wouldn’t need to handle the fish at all. If others find this idea intriguing, I’d love to get data for fish of different varieties and sizes to validate the claim.