There's less and less reason to do it (and it's never 5). On systems without floating point you might want to round it a bit, but only if the specific thing you're doing allows it, and even then you're more likely to do a fixed-point approach by using e.g. 314 and dividing by 100 later, or adjusting that value a bit so you can divide by 128 via bitshift if you're on a chip where division is expensive. However, in 2025 you almost certainly should have picked a chip with an FPU if you're doing trigonometry.

And while rounding pi to 3 or 4 is certainly just a meme, there are other approximations which are used, like small-angle approximations, where things like sin(x) can be simplified to just x for a sufficiently small x.

For back-of-the-envelope or mental calculations, pi is often 3 or 10^(1/2).

The latter is better than 1% accurate, and has nice properties when doing order-of-magnitude/log space calculations in base 10.

you've never seen anybody simplify it to 3 when doing head calcs without a phone nearby?

it doesn't happen often, in fact I've seen it once. in a decade.

There's a lot of weird stereotypes out there that make no sense. Like the whole "programmers wear thigh high socks" thing. Where did that even come from?