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When I was in 6th grade (U.S. so 12-13 years old), I took a summer school class. The teacher gave us all different sized spools (spools that hold sewing thread but were empty). We each made a mark on one wheel then rolled it one revolution over paper and marked how far it went (i.e. one circumference). Then we held the spool on its side and measured how may diameters it took to cover that distance. And, no matter what size the spool, it took a little over 3 diameters.

In looking back this was a great exercise to prove that circumference equals $\pi$ * diameter. I was wondering if this demonstration is common.

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    $\begingroup$ I have my college students measure across and around a circle, so they can see this. Many of them don't really get it. (And some of them give me measurements that give pi perfectly. I tell them they were supposed to measure, because I know those didn't.) $\endgroup$
    – Sue VanHattum
    Commented Feb 23, 2023 at 5:15
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    $\begingroup$ I use basically this same demonstration (community college arithmetic), and it is interesting how many students who have memorized "2 pi r" are surprised by the physical demonstration. Some of them, of course, don't immediately see the connection between $2 \pi r$ and $\pi d$. But when we're counting how many diameters it takes to make the circumference, and we get to two and I ask "how many is this going to take", they almost all say "three". Clearly they're seeing something they don't expect, even after memorizing and using the well-known formula. Seeing $\pi$ like this is novel for them. $\endgroup$
    – Nick C
    Commented Feb 23, 2023 at 17:48
  • $\begingroup$ @NickC Connection? d=2r by definition. So it makes three diameters, not sure what "two" you were talking about. $\endgroup$
    – Rusty Core
    Commented Mar 1, 2023 at 16:31
  • $\begingroup$ @RustyCore I guess I'm not sure what "two" you're referring to here. Do you mean when I say that we're "counting how many diameters it takes...we get to two" ? $\endgroup$
    – Nick C
    Commented Mar 1, 2023 at 16:51

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I don't know how common this demonstration is (I never saw it in a class when I was a student), but there's a nice video illustrating it that students can watch: see the first 35 seconds of the Vertasium video on pi: https://www.youtube.com/watch?v=gMlf1ELvRzc.

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