When the approximation $\pi\simeq 3.14$ is NOT sufficent

It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example of such a purpose, accessible to a middle school student.

Thanks.

If you want to calculate the volume of an $n$ dimensional sphere with radius $r$ you must include a power of $\pi^{n/2}$.

In general, we have $$V_n(r) = \dfrac{n\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^{n-1}$$

If we consider the ratio $R_n$ of the actual volume to the approximate volume we find that $$R_n = \frac{V_n(r)}{V_n(r)_{\text{approx}}} = \left(\dfrac{\pi}{\pi_{\text{approx}}}\right)^{n/2}$$

If we set $n = 2\cdot 40^{20}$ we find that $R_n \approx 1.10621$ rather than the expected value of $R_n = 1$.

This example is completely contrived but it demonstrates that the error will become apparent when large powers are involved. This is just a simple application of the use of significant digits.

There are many more examples using periodic functions with a period involving $\pi$. Consider finding

$$\tan(10^{100})$$

Essentially, you must find $10^{100}\! \pmod{\pi}$ which would require knowing $100$ digits of $\pi$ (to calculate the answer with some precision). You can find more information here. Again, this involves large powers requiring an excessive number of digits.

The easiest example of the need for an exact value most likely comes from numbers of an astronomical (or even extra-astronomical) scale. I don't know how convincing this argument would be to students and you can consider cross posting to MathEducators.

If you want an explanation for younger mathematicians you should explain how large powers of any number can an estimate unreliable and then present an applications of large powers of numbers or even just large powers of $\pi$. You can show how estimates like $\frac{1}{3} \approx .333$ are unreliable and then move up to more complicated examples.

The BBP formula, a spigot algorithm for $\pi$, required knowing the value of $\pi$ to high precision in order to use the integer-relation algorithm which generated the formula. Perhaps the circularity (knowing $\pi$ to find a formula for $\pi$) makes this less interesting, but in general integer relation algorithms are one of the best reasons for caring about extremely high precision in constants.

You could tell a middle school student:

32 digits of pi is enough to calculate the circumference of the visible universe to within an error invisible under a light microscope.

Be prepared to acknowledge that 32 digits of pi is therefore more than enough for "practical" purposes, while two digits of pi is often (but not always) satisfactory precision.

Dynamical systems. E.g. the earth, sun plus a communications satelite system: this is a classical example of a 3-body problem where the mass of the satelite doesn't affect the earth-sun system, who follow an elliptical trajectory (2-body problem), but the earth and sun do affect the satellite. The trajectory of the satelite, however, doesn't have an analytical expression combination of known functions, and numerical methods and power expansion is needed to predict it. The number $\pi$ appears in some expressions of the series, and it is necessary to have a lot of decimals as the error may increase exponentially and the satellite crash.

Of course in real satelites the trajectory is analyzed and may be corrected. Nonetheless, high precision is required.