The Mishna in Ohaloth 12:6 informs us that you need a Tefach of width to bring Tumah from one location to another:
קוֹרָה שֶׁהִיא נְתוּנָה מִכֹּתֶל לְכֹתֶל וְטֻמְאָה תַחְתֶּיהָ, אִם יֶשׁ בָּהּ פּוֹתֵחַ טֶפַח, מְבִיאָה אֶת הַטֻּמְאָה תַחַת כֻּלָּהּ. וְאִם לָאו, טֻמְאָה בוֹקַעַת וְעוֹלָה, בּוֹקַעַת וְיוֹרָדֶת. כַּמָּה יִהְיֶה בְהֶקֵּפָהּ וִיהֵא בָהּ פּוֹתֵחַ טֶפַח. בִּזְמַן שֶׁהִיא עֲגֻלָּה, הֶקֵּפָהּ שְׁלשָׁה טְפָחִים. בִּזְמַן שֶׁהִיא מְרֻבַּעַת, אַרְבָּעָה, שֶׁהַמְרֻבָּע יָתֵר עַל הֶעָגוֹל רְבִיעַ:
[With regard to] a beam which is placed across from one wall to another and which has uncleanness beneath it: If it is one handbreadth wide, it conveys uncleanness to everything beneath it; If it is not [one handbreadth wide], the uncleanness cleaves upwards and downwards. How much must its circumference be so that its width should be one handbreadth? If it is round, its circumference must be three handbreadths; If square, four handbreadths, since a square has a [circumference] one quarter greater than [that of] a circle.
Firstly, regarding the circle, as we all know, if the circumference is only 3 tefachim then we are missing about 3mm from the required Tefach.
But more peculiar is the "divide the circumference of the square by 4". It seems one could simply measure any one side.
However, even that isn't accurate, for if the square beam isn't sitting perfect horizontal - like this: ■ - but is (in the extreme case) sitting tilted - like this: ♦ - then the "divide the circumference by 4" is totally wrong! We now have 1.41 Tefachim. A circumference of 2.84 Tefachim would be sufficient to get us a 1 Tefach for the Tumah to travel underneath.
So: Why is the Mishna assuming the beam is sitting "square" (like this: ■) and why do the meforshim all seem perplexed by the unnecessary mathematics, but none of them seem to notice that we have a real question if the beam is sitting like this: ♦ ?