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I am about to do some calculations involving Pi. These would be easy to do in a spreadsheet, but I need the result to use Pi as a symbol - not as an approximated value.

Eg:

   A   B    C
1  =5  =Pi  =A1*B1
2  =2  =3.2 =A2+B2*C1*Pi

Should give:

   A   B    C
1  5   Pi   5*Pi
2  2   3.2  2+16*Pi^2

Does such a spreadsheet exist? I have the feeling that there will be so few tools that I cannot really be picky about the OS or price. If there are many of such tools, I would prefer GNU/Linux and free software.

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I think you should be looking at computer algebra system software or symbolic algebra systems. Wikipedia has an extensive list of options. Few if any use a spreadsheet metaphor. Most are more notebook based. The big names are Mathematica and Maple, both of which are expensive. There are a lot of open source options such as Maxima and SageMath. Computer algebra systems can do more than just work with irrational numbers, they can solve equations and even do calculus symbolically. This Mathematics SE question may have useful information. Wolfram Alpha can do some of this for you on the web. For example: enter image description here

If you desperately want to stay with the spreadsheet metaphor, then this paper might be of interest. Otherwise if you have some Python programming skills you could try Pyspread which is a Python based spreadsheet that allows Python expressions in cells. In conjunction with SymPy, you chould get what you are looking for, but I haven't tried it myself.

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