Suppose that the shape of a fish pond is irregular, so we can't use any simple math formula, e.g. width $\times$ length $\times$ depth, to find its volume.

I'm wondering if, there's a good way of estimating its volume? For example, one can estimate the size of a molecule using an oil film. Reversely, one can also estimate the size of the surface knowing the oil molecule size. Could there be a similar way to measure not the surface area, but the volume?

It's not a real project, but an interview question.


closed as off-topic by Bill N, Jon Custer, Kyle Kanos, Brian Moths, Yashas Sep 7 '17 at 16:01

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    $\begingroup$ Perhaps they just wanted an back-of-the-envelope estimate, not an exact measurement? $\endgroup$ – Deep Sep 6 '17 at 5:07
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    $\begingroup$ you could treat the fish as harmonic oscillators $\endgroup$ – Señor O Sep 6 '17 at 6:06
  • $\begingroup$ @Deep yeah, a simple and sweet one.. $\endgroup$ – athos Sep 6 '17 at 6:13
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    $\begingroup$ If "irregular" is the only infomation about the fish pond, then the question isn't really about giving an answer but about hearing how you approach the problem, i.e., they want to know how you think about problems and solutions. $\endgroup$ – Bill N Sep 6 '17 at 13:23
  • $\begingroup$ Seems to me that this is effectively a list-based question, which is generally considered off-topic as too broad (because just about every response is a valid answer). $\endgroup$ – Kyle Kanos Sep 7 '17 at 10:08

Probably the most accurate way is to use a radioactive tracer as is used to measure blood volume but . . . . . . .

Or you use some inert chemical compound.

Look up effect on pond life, both plant and animal) of a low concentration of common salt in pond water.
A concentration of less than 5 parts per thousand ($5\, \rm ppt$) should be perfectly fine even for koi carp and indeed this is a method which is used to kill harmful parasites.

Measure the salinity of the water in your pool $S_{\rm initial}$ using a conductivity/salinity meter of which there are numerous types.

Estimate volume of pond in litres.
Add 2 kg of common salt (sodium chloride) per 1000 litres of water (total $M$ kg) which will increase the salinity of your pond by about $2\, \rm ppt$. Less/more salt will produce less/more accurate results. Best to use salt purchased from a koi/fish shop to ensure it is pure and without any potentially toxic additives.


Measure the new salinity of your pool $S_{\rm final}$

Volume of pool $\approx \dfrac{M*1000}{S_{\rm final} - S_{\rm initial}} $ litres.

Another way is to take known volumes of water before and after the addition of the salt, allow the water to evaporate, find the difference in weight and hence evaluate the volume of the pond.

  • $\begingroup$ I wonder if it would be less harmful to use a weak acid and measure the pH change. Many acids can be easily neutralized, while salt will stay in the pond for a long time. $\endgroup$ – Dmitry Grigoryev Sep 6 '17 at 12:38
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    $\begingroup$ @DmitryGrigoryev I have no experience of doing that. A possible problem is that you might have to add fairly concentrated acid to get a measurable change in the pH. $\endgroup$ – Farcher Sep 6 '17 at 14:56
  • $\begingroup$ pH is a log number, as i recall during middle school chemistry class, near pH=7 it's hard to measure accurately.. $\endgroup$ – athos Sep 6 '17 at 23:04

For a fairly exact measurement, I would take some harmless chemical that's easy to detect, and pour a certain volume in and let it dissolve/diffuse completely. Then, measure it in ppm, do the necessary calculations, and get your volume.

  • $\begingroup$ I have a similar idea. What chemical do you have in mind? Salt? That could be measured by boiling etc. But it might not be harmless $\endgroup$ – innisfree Sep 6 '17 at 6:13
  • $\begingroup$ should i transfer it to chemistry board... $\endgroup$ – athos Sep 6 '17 at 6:15
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    $\begingroup$ No, I think it's interesting. $\endgroup$ – innisfree Sep 6 '17 at 6:19
  • $\begingroup$ One could use a bio-compatible pigment (like the one they use to study sea currents) and find out its final concentration by measuring the turbidity, provided the water is initially clean enough. $\endgroup$ – Dmitry Grigoryev Sep 6 '17 at 12:25

I propose

1) emptying the pond and putting the fish in a storage container: Usually there is a built in container at the bottom of the pool (for when it needs to be cleaned), where the fish go when it is emptied, it can be the storage container since you will not be adding cleaners.

2) measuring the rate of flow of water that will fill the pond: take a 20 littre bucket and measure the time it takes to fill up completely. Doing it for a few times will give you an estimate of Δ(time)

3) Time how long it takes to fill the pond.

4)divide the time by the time of twenty litres. Multiply the number by 20. This will give you the volume of the pond because 1 litre of water fills 1000 cc .

You can estimate your error by using analogously the Δ(t).

5)put the fish back in (or they will swim out of the hole by themselves while it is filling, then their volume will add to the error , but it will not be a large number..

  • $\begingroup$ That's a whole lot of water to waste for the measurement, though! $\endgroup$ – innisfree Sep 6 '17 at 6:35
  • $\begingroup$ @innisfree do it when the water has to be changed? they do clean fish ponds. Also before filling the first time will not waste water. the bottom container which is usually a bucket has to be taken into account. $\endgroup$ – anna v Sep 6 '17 at 6:37
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    $\begingroup$ fishes will not be happy... $\endgroup$ – athos Sep 6 '17 at 9:50
  • $\begingroup$ Good in principle but not as good in practice. A pond might have more animal than fish in it. The "mud" at the bottom can cause difficulties. Where does the water come from? If from the tap then it needs to stand for a period of days to allow the chlorine to disperse. $\endgroup$ – Farcher Sep 6 '17 at 11:32
  • $\begingroup$ @Farcher I am assuming a constructed fish pond. I guess coming from a dry country those are the only ones I know. There are in the mountains lakes with fish, but no fish ponds. $\endgroup$ – anna v Sep 6 '17 at 11:50

If you can measure (or reliably estimate) the area taken by the pond, you can measure depth in several uniformly distributed random points and calculate the volume as area × mean depth. As you keep increasing the number of depth measures, this estimate will converge towards the true value of the volume.


I propose diluting the pond to measure its volume. The liquid added could be e.g., water of another temperature or a coloured die that dissolves into the pond. Let's take water of another temperature,

  1. Measure the temperature of the pond
  2. Add a known volume of water at a known temperature to the pond
  3. Wait a short time
  4. Remeasure temperature of the pond

From the final temperature, initial temperature and volume of water added, you can infer the volume of the pond. I believe the relation for adding hot water, $T_{added} \gg T_{pond}$, is $$ V_{pond} \approx \frac{T_{added}}{\Delta T_{pond}} V_{added} $$ where $T$ is in Kelvin (or a similar unit with an absolute zero).

My brief research on this topic suggests that this is a genuine problem in fish pond management, as one must add chemicals per unit volume of water in the pond. I see that dissolving an innocuous chemical into the water is indeed the preferred technique.

My method is conceptually clear but probably impractical, as it might be difficult to change the temperature of a big pond by an amount that could be measured. Probably an analogous technique with some other chemical would be better in general, though might require more sophisticated equipment. I can't think of any chemical that is easy to measure and harmless (e.g., salt or acidity could be readily measured but may be harmful). Maybe a colored die.

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    $\begingroup$ this doesn't sounds right as the air and pond will have heat transmission. $\endgroup$ – athos Sep 6 '17 at 9:49
  • $\begingroup$ Right, it's just an approximatexample method $\endgroup$ – innisfree Sep 6 '17 at 10:04

Take an integral of the surface of the bottom of the pond over the height of the pond at any point. This demonstrates your knowledge of multivariable calculus as it involves taking a 2 dimensional integral over 2 variables.

  • $\begingroup$ measuring that might not be simple in practice? $\endgroup$ – athos Sep 6 '17 at 4:38
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    $\begingroup$ good call cause most ponds have an analytic function at the bottom. $\endgroup$ – Señor O Sep 6 '17 at 5:59
  • $\begingroup$ @SeñorO who said it had to be an analytic integral. Simple numerical calculations from measured depths would be enough. In fact, this is the only way to measure volumes of bodies of water in the real world. The methods described in the other answers are overly complex and require unnecessarily exact measurements. $\endgroup$ – Johnathan Gross Sep 6 '17 at 6:11
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    $\begingroup$ I suspect that the point of the interview question is to make a candidate think of innovative experimental ways of measuring a volume. Just doing the integral is IMHO a silly answer, that is not in the spirit of the question. $\endgroup$ – innisfree Sep 6 '17 at 6:20

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