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I've been looking around google trying find an answer to this but because i can't i would like to ask here for even a basic answer, and maybe an explanation.

I am wondering if in any way to figure out the volume of any given three dimensional shape, which follows absolutely no "rules", such as this shape here such as this shape here

A lot of the explanations i've seen, assumes one side of the object to be at a fixed size/shape, much like this thread here: Volume of irregular solid

While others calls the examples an "irregular shape", but really its just a shape composed of multiple basic shapes

But for shapes that follows absolutely no real logic to it, is it possible to calculate the volume of such an object? And is it possible to do this without needing to break it down to "multiple basic shapes"?

If not, could some one provide an explanation as to why? I was reading something called a Heron's Formula to calculate ANY triangle's area so long as all sides are known. Is there not really an equivalent to calculating a volume of any given shape?

Also, is it possible to some how take the surface area of a random shape (that is, all the areas of each triangle that composes 3d shape and adding them all together), and some how convert that into volume? If not, why is this the case? Could someone provide an example of two shapes having the same surface area but different volume?

user3696118
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Given enough information (e.g. area, angle, edge lengths) about the sides, it is possible to figure out the volume of a polyhedron. But the formula could be much more complex. Heron's formula for the volume of a tetrahedron is an example.

A triangle is determined by its side lengths because of SSS theorem. But for random polyhedron, knowing the area of each side is obviously not sufficient. And you may also find out that a parallelepiped is not determined even all of its edge lengths are given. In this sense, calculation on the volume of a random solid with curved edges is way harder, and is impossible to do with only all the areas known. So breaking polyhedra into basic shapes might be one of the most convenient methods.

Eric Monlye
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