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I am looking to make a triangular prism with some special properties but with no idea how to go about doing the calculations. The idea is to have the volume above each face of a triangular prism be equal if the prism was put into the smallest possible sphere.

I have taken calculus 2 and done a bit of work with derivatives and integrals, however this seems like it would require triple integration and polar coordinates, neither of which I know how to properly use.

Here is an image of the area above a surface for a cube (easier to draw a cube).

enter image description here

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    I don't think I can visualize "the volume above each face" without more information. Can you draw a picture? – 2017-01-26
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    Edited the question with a image. The one in the top right is probably the best, but each could work. – 2017-01-26
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    Understood. The vertices of the prism will be on the sphere. The triangle will be equilateral. If you don't get an answer from someone here in a day or so I might try it. – 2017-01-26
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    It's hard to understand what you mean. Do you want your triangular prism be "projected" onto its circumscribed sphere as if there was a light bulb in the center of the sphere ? – 2017-01-26
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    @JeanMarie yes. Except in this case it would be the volume of the shadow, not just the area of it on the inside of the sphere – 2017-01-27
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    @Ethan Bolker Not quite, the triangle wont necessarily be equilateral. The goal is to adjust the length of the prism to change the volumes such that the volumes above each face are equal. – 2017-01-27
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    @JonathonM I think the triangle will have to be equilateral so that the volumes "above" the rectangular faces can be equal to each other, as well as to the volume above each triangular face. There's clearly a unique height to base edge ratio that works. – 2017-01-27
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    Have a look at the figure I have drawn (using Powerpoint...). – 2017-01-27
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    @Ethan Bolker yes, sorry. I was thinking of another thing when you said equilateral triangle. It will be an equilateral triangle. – 2017-01-27

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This is not an answer: it is the only way I have found to display a figure.

@Jonathon M Could you explain with the notations of the figure below what you exactly desire ? Is it the numerical value of the volume of ABCA'B'C' ? And what for (it can be interesting for us to explain your ultimate goal).

enter image description here

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    The goal would be to keep AC, AB, and CB the same lengths. Then adjust the height of the triangle so that the volume of the object consisting of the faces: (A'B'C'),(A'B'AB),(B'C'BC),(A'C'AC),(ABC) would all be equal to each other. The resulting answer would be a ratio of a length AB to the height of the prism. The purpose of this is for a die that is mathematically perfect, albeit physically lacking. – 2017-01-27
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    I see: a well equilibrated die with 5 faces. – 2017-01-27
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    Why does making those volumes equal make the die "perfect" - assuming you mean equally likely to come up any of the five ways? Do you mean volume as in your question, or the area of the region on the surface? Is the physics well known? – 2017-01-27
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    @Ethan Bolker the physics is well known, and it is well known to not work in this scenario. There are too many variables to account for to make a physically perfect 5 sided die. However, eliminating the pesky reality, you are left with this method of finding probabilities of each face. It is the volume of the area above the face until it reaches the sphere, not the projected surface area of the sphere. – 2017-01-27
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    Some thoughts: 1) As a square is the union of two triangles, you need only look for volumes of the type given in my figure 2) An efficient approximate method for computing volumes is MonteCarlo simulations 3) You will find formulas for computing the volume of the portion of sphere OA'B'C' (formula to be found by calculation or most probably in tables) to which you will substract the volume of OABC (which is $(1/6) det(\vec{OA},\vec{OB},\vec{OC})$) – 2017-01-27