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Assume a physical object with arbitrary shape whose dimensions are on the order of 1 meter. How would one measure the surface area of the object?

There may be many reasons for determining the surface area:

  • Plating or painting the object
  • Determining rate of sublimation
  • Determining how strongly it smells
  • Determining heat loss

If more detail is needed, then assume the object is a rock or stone.

This question is inspired by a similar controversial question on Physics.SE. In that thread it was clear that those with a mathematical background felt that the question was ill-defined, yet some creative answers were proposed. As this is really an engineering question, I feel that a generalized, better-defined version of the question would be on-topic here. See also the meta thread discussing the merits of asking this similar question on Engineering.SE.

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  • $\begingroup$ Why is it an "engineering" question - it applies to many disciplines... $\endgroup$ – Solar Mike Nov 12 '19 at 12:31
  • $\begingroup$ It has some very good answers: physics.stackexchange.com/q/512834 $\endgroup$ – Solar Mike Nov 12 '19 at 12:32
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    $\begingroup$ I can't think of a better answer than the paint weighing ideas discussed in the Physics answers. Dip once to fill in the cracks with a paint of low viscosity according to how smooth you want it to be, and wait to dry. You then weigh, dip, and weigh again, compared against a known surface area reference object dipped at the same time. EDIT - I just spotted that your object is around 1 meter, so may be impractical to dip. What about 3D scanning or Photogrammetry, then measure the surface area of the 3D Mesh? $\endgroup$ – Jonathan R Swift Nov 12 '19 at 14:23
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    $\begingroup$ @SolarMike: This is an engineering question as this is a problem without an obvious solution, which for any practical purpose has an answer but for which the mathematical models are inadequate to answer. $\endgroup$ – dotancohen Nov 12 '19 at 14:47
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    $\begingroup$ So, it's an engineering question. It's a complicated engineering question. When you have a complicated engineering question with a bunch of possible answers to choose from, you pick the answer that is best for the situation. It may be a different answer from the one you picked yesterday, and the one that you pick tomorrow may be different, but today you pick one and run with it. Since we don't know your specific situation, how can we answer? $\endgroup$ – TimWescott Nov 12 '19 at 15:49
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Photogrametry usually works quite well unless your object has some very complicated internal cavities. The good thing about photogrametey is that you can easily collect the data with any decent smartphone. Just remember that cameras lose scale so be sure to survey atleast one dimensions with a measuring tool. Or instrument the scan with a known object. There are lots of photogrametry packages even online ones that can triangulate for you. Then just open it in a 3D app eliminate extra faces and count the surface area.

If you want more fancy methods you can use structured light scanners lidars and so on. Hell you can even use 6-12 pictures and milk.

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  • $\begingroup$ This is great. It seems that Meshroom can convert a series of photographs to a mesh file, and Blender can then calculate the area. Both are open source and run on Linux. $\endgroup$ – dotancohen Nov 12 '19 at 21:41
  • $\begingroup$ N.B. The output from Meshroom is not scaled - so you will need to take a measurement of the object in real life and scale the model in software to match it. $\endgroup$ – Jonathan R Swift Nov 13 '19 at 17:31
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May sound not very precise but i take a stab on it.

Wrap your object with wet thin paper, engineering tracing paper for example(Bienfang paper), let it dry with a hair dryer, then cut it into flat squares and triangles.

Measure and add the surface area of the pieces.

Alternately you can project a narrow beam of light so that it cast shadows of many angles on a screen which you can measure the main axis off the shadows and reassemble the geometry including the surface and volume.

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If budget is not an issue, there are 3D scanners with ‎0.1 mm of resolution.

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  • $\begingroup$ Only 0.1 mm? Hell even my self built laser line scaners can do better than that. $\endgroup$ – joojaa Nov 13 '19 at 18:27

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