Realistically, the difference between a Euclidean-geometry sphere and a Chebyshev-geometry sphere really isn't as big as a lot of folks think--and, more importantly, the difference between a Chebyshev-geometry circle and a Euclidean-geometry circle is smaller still. With the way we approximate these things, you only get a meaningful difference (that is, more than about +/- 10%) when you get out to radii of about 35 feet. That is, because we approximate these things with chunky squares regardless,
People throw around terms like "square fireballs" or "firecubes" without actually putting any thought into what they're complaining about, most of the time. "It's utterly ridiculous to have a square-shaped fireBALL!" Okay, but we already have fire"ball"s made of cubical chunks, not an actual sphere to begin with. (Or, rather, 99.9% of the time, we have fire"circle"s made up of a bunch of chunky squares.)
We can quantify the difference between the 4e approximation--aka Chebyshev geometry--and a
true circle quite easily. It is, always and exactly, (4-π)/π, or about 27.324%. But the difference between a chunky-approximated circle and the 4e approximation is rather less clean, because it's not going to be a perfect amount and it's going to vary as our circles get larger--but, crucially, the chunky-approximated circle is
also wrong, albeit often by a smaller degree.
As an example: Imagine a circle of radius 15. Per 5e's rules, that circle must be centered at a grid intersection point, and will
entirely cover every 5' square which has at least 50% of the area inside the circle. As a result of this, for a circle of radius 15'...the only difference between this "circle" and a 4e-style circle is
four corner squares. Indeed, the same applies for a 10' circle as wel. Here's a table for radii from 5' up to 40':
| Radius (ft) | Diff (5' sq.) | Relative diff (%) |
| 5 | 0 | 0 |
| 10 | 4 | 33.33...% |
| 15 | 4 | 12.5% |
| 20 | 12 | 23.076923...% |
| 25 | 20 | 25% |
| 30 | 32 | 28.571428...% |
| 35 | 40 | ~25.64% |
| 40 | 48 | 23.076923...% |
As you can see, it's dancing around the 27% figure, depending on whether the base circle is an under-estimate (as is the case for 15' and 30') or an over-estimate (as is the case for most circles). While the absolute difference obviously rises without bound, the
relative difference is...about a quarter.
So people are literally freaking out over this so-called wildly unrealistic approximation...that basically just increases areas by about 25% relative to the approximations we were already using. Indeed, in most cases, the error is actually
smaller than 25%, because the "at least 50% covered" thing is also usually an over-estimate, albeit not always.
