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I know there are a lot of ways to calculate $\pi$. But is there a way to explain and calculate it together with a kid, using only basic math and geometry? So without trigonometric functions, square root, etc.?

Edit: sorry, maybe the question is confusing. I am aware of how to approximate the value by measuring inner and outer polygons. The question is really about getting close to $\pi$ with calculations and not measuring.

vilmarci
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    The simplest way to demonstrate to a kid will always be with a circle – Stephen Donovan May 27 '21 at 06:46
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    No square roots excludes Pythagoras, which is fairly basic math, and I don't think you can go too far towards $\pi$ without it. You could calculate it with Buffon's needle without any math, but would have a hard time trying to explain why that works. – dxiv May 27 '21 at 06:47
  • Or you could compute it, without trig or square roots, using a spigot algorithm, but good luck trying to explain how that works, too! – Hans Lundmark May 27 '21 at 07:41
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    Generate zillions of independent uniform random numbers in the interval $[-1,1]$. Group them in pairs representing $(x,y)$ coordinates. Calculate the proportion of these points that satisfy $x^2 + y^2 \leq 1$. That proportion will be $\pi/4$ in the limit. –  May 27 '21 at 08:13
  • $cos(\theta) = 0$ has a unique solution between 1 and 2, as you can see from its graph. The Bisection Method will let you calculate $\theta$, which has value $\frac{\pi}{2}$. You can relate this to the x value on the unit circle at angle $\frac{\pi}{2}$. So cosine graph, Bisection, unit circle - maybe still too much? – Paul May 27 '21 at 16:59

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Draw as close to a perfect circle as you can with a compass. Measure it's perimeter or area and divide by the diameter or the radius squared, respectively, to compute $\pi.$

To measure perimeter, try wrapping a string around the circle and cutting it. To measure the area, try cutting out the circle and weighing the piece of paper you printed on it. Or you could be like Archimedes if your kid knows some geometry: Compare the circle with an inscribed and circumscribed polygon, which hopefully you can calculate the area of more easily, to get an upper and lower bound.

  • Thank you for the effort and reply, I updated the question. Measuring is ok, I was looking for calculation ideas. – vilmarci May 27 '21 at 10:28
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Get a piece of string and a tape measure. Measure the circumference and diameter of as many circular objects as you can find (tins, plant pots, ...). The ratio of the two is then $\pi$. Draw a graph of circumference against diameter if they are a bit older. The points will lie on an approximate straight line and the slope is a good measure of $\pi$.

Paul
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  • Thank you for the effort and reply, I updated the question. Measuring is ok, I was looking for calculation ideas. – vilmarci May 27 '21 at 10:29