Work in progress...

• The math for the "trench" carve radius assumes that the board has zero taper.
• The "trench" carve radius tightens up a lot more than I expected, so I'm a little skeptical.
• The math to calculate the speed for a balanced carve on deformable (trenched) snow has not yet been implemented.

Some relationships to bear in mind...

Obvious

For a given edge angle, a board with a shorter sidecut radius will (as you might expect) carve a tighter turn than a board with a longer sidecut radius.

For a given sidecut radius and carve radius, there is only one edge angle that will make that board carve that radius.

You can't carve a turn whose radius is larger than your board's sidecut radius. You can follow the path of such a turn, but you'll be skidding, not carving.

If you ride a few boards with different radii, you'll find that shorter radius boards are best suited for lower speeds, and larger radius boards are best suited at higher speeds.

• Shorter radius boards carve tighter paths, and the faster you go, the more awkward it gets to carve tight turns - it's just easier to make bigger turns, which require low edge angles, and carving becomes impossible because the centripetal force pushes you out of the groove.
• Longer radius boards carve larger-radius paths, so the slower you go, the more awkward that gets. The board won't change direction quickly enough unless you use a very high edge angle, and at low speeds that just means you'll be riding the sidewall. So it's just easier to skid instead.

So, if you're having trouble carving at higher speeds - or if you find yourself subconsciously slowing down before you start carving - a larger sidecut radius might be for you.

Less Obvious

For a given carve radius, there is a pretty narrow range of speeds that will let you carve that path.

• Go too slow, and you'll be riding the sidewall, not the base. Or you will fall to the inside.
• Go too fast, and the board will make the turn, but your body won't.
• You can cheat somewhat with angulation and skill, but you can only cheat for so long before the laws of physics catch up to you. You have a lot of flexiblity for quick course corrections, especially if you don't mind transitioning between carves and skids. But if you want to make sustained arcs? Newton's laws rule.

If you want to make highly inclined carves where your shoulders are skimming the snow, you might think that a tighter sidecut radius would be better. It turns more tighter and more tighter is more better, right? Not necessarily. When you lean way into a turn, with a tight sidecut, the board whips around in a very tight arc. With a 10m sidecut, it feels like my head stays still and my feet rotate around me. (That's not what really happens. I am exaggerating. But it really does feel that way.) With 13m or 15m sidecut radius, the arc is much larger. It feels more like snowboarding and less like acrobatics.

If you'd like to know more about the math behind the calculator, see Physics of a carved snowboard turn.

For even more math, check out balanced carving turns in alpine skiing or Alpine Ski Motion Characteristics in Slalom (which is where I got the math for the "trench" carve radius).

If you find yourself thinking that sidecut depth and effective edge length matter more than sidecut radius, see Appendix A of the "balanced carving turns" paper. Spoiler: The math works out the same.

If you wonder about the accuracy of Howe's cosine formula, consider this quote from the 2nd paper:

...in the field study of turns performed by members of the Norwegian national team, Reid, Haugen, Gilgien, Kipp, and Smith (2020) found a good agreement with equation (33) up to Ψ ≃ 70◦

"Equation 33" is Howe's cosine formula, and Ψ is just the edge angle.

You may have heard of an attempt to derive carve radius using finite-element-analysis. As noted above, it didn't align well with reality.

A lot of people greatly over-estimate the role of board stiffness and/or effective edge length in determining sidecut radius. Those parameters do affect how well a board can hold a carve, but they have negligible effect on carve radius.

I own two snowboards that are 170cm long and have 13m sidecuts, but the effective edge on one of them is very short - it's a powder board, and the tip and tail start to curve upward just past the bindings. Edge hold is not great on the powder board, but it's still capable of hip-dragging carves when the snow is good, and both boards carve the same paths.

To give you a very rough idea of what boards tend to have what sidecuts...

• Freeride snowboards for soft-boot riding typically have a sidecut radius under 10 meters. This makes them agile at low speeds, but twitchy at high speeds.
• Slalom hard-boot snowboards were typically around 10 meters.
• Boardercross snowboards typically range from 10-13 meters or so. If you're looking for a soft-boot board that can carve big turns, look into this category - but look at the specs closely because they vary a lot.
• Giant slalom snowboards are typically around 15-17 meters, however I've been seeing them with sidecuts of 18-20 meters lately.

The generalizations above are just generalizations.
All generalizations are false.
Hopefully those generalizations are helpful.
Check the specs before you buy.