What are some commonly used containers that have peculiar exact volumes?

Question

I am doing a project in calculus in which I must calculate the volume of some kind of container. I didn't wish to choose a boring object. So I wanted to examine the options from the extensive mathematical community. What are some common container-type items (such as a soda can or Gatorade bottle) whose exact volume has interesting properties? For example, suppose the object's volume must be derived via an infinite series or involves the constant e. Or suppose the designers of the product chose, for some reason, to make their container have some kind of interesting or peculiar volume property. Ideally I would prefer the container to be easily able to attain or which is commonly used by the public. Please provide what the object is and what it's exact volume formula is.

Answer 1

Any real-world object can be non-boring if you look to find its volume in enough detail.

You can approximate a basketball as a sphere. But what about the shallow channels on the ball? How long are they? How would you approximate the volume they carve out of the sphere? Or all of the little bumps on the ball: How much volume do they add?

Or a can of soup. Yeah, it's mostly cylinder, but you have the seam of the can, and the caps, and the ridges around the circumference.

Etc., etc., ...

Answer 2

What are some common container-type (such as a soda can or Gatorade bottle) whose exact volume has interesting properties?

That highly depends on what is interesting for you.

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E.g. a gas storage here in Germany is often a big ball of metal, because it fits a maximum of volume for a given surface area of steel and probably distributes stress very evenly.

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A container like the small glass Coke bottle seems to be modeled after the female body ("Mae West bottle") to please aesthetically, but the official reason is that it had to be distinctive from the bottles of the competition (source).

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Competitor Pepsi seems to use the golden ratio in its product design.

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A container for soy sauce, like the Kikkoman dispenser by Kenji Ekuan looks Japanese, a bit like a torii gate.

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Standard containers are the backbone of globalization, now and then.

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The most prominent container in computer graphics might be the Utah teapot. It acts as reference model, e.g. to test various rendering algorithms.

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Physics has some iconic containers as well, like the Leiden jar.

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I want to finish with the Klein bottle. Alas it might give you some headaches. :-)