RAW yes, they're 30 feet away.
As a home rule, I'll sometimes run total distance = long distance plus half the short distance. That also correlates nicely with making every other diagonal count as 10'
this post was submitted on 10 Jun 2025
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But D&D uses Chebyshev distance, not Euclidean. No need for Pythagoras. And Pathfinder alternates between Chebyshev and Manhattan to approximate Euclidean.
Me at 20: I'm never going to need Chebyshev distance in real life. Why am I learning this?
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Happy cake day to you!
Thanks! Quite a few of us seem to have our cake days around this time...
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Ah yes, the "APIcalypse" or "Rexxit".
Playing combat on a grid is actually presented as an optional rule and not the default for 5E, despite its popularity
So are feats, and point buy.
And multiclassing.
Of those that has been the least common at my tables.
i wish that it was more common to refer to the metrics in terms of what they are instead of who discovered them. i can’t ever remember off the top of my head if the chebyshev one is supposed to be the diamond metric (L^1^) or the square metric (L^∞^).
Personally I find Euclidean easy to remember because it matches the much more general Euclidean geometry. So you just remember "this is like, real maths". Manhattan distance is easy to remember because it does basically "refer to the metrics in terms of what they are", so long as you remember that Manhattan famously is a grid. Chebyshev is the hardest, but for me it's a simple matter of "the one that's left over".
I have no idea, based on the name, what diamond and square metrics are supposed to be.
i think that’s a good point and that is a nice way to remember them. i think a lot of it just comes down to personal preference.
i like calling them the diamond/square/circle metrics because those shapes describe the sets of points that have unit length. i’ve found this wikipedia picture to be very helpful, and the diamond/square/circle terminology is my way of paying my respects to the picture.
Ah right, so "diamond" (depicted as a square rotated 45 degrees) is Manhattan, circle is Euclidean, and square is Chebyshev, then?
yeah exactly. i understand it as follows:
- in the manhattan metric, points have length one if the lengths of their coordinates sum to 1. so you get the points (1, 0), (0, 1), (-1, 0), and (-1, -1). and then you connect these four points with straight lines to get the diamond shape. this follows from the observation that if the x coordinate decreases in length by 0.1, then the y coordinate must increase in length by 0.1.
- in the euclidean metric, the points of length one lie on the unit circle, since x^2^ + y^2^ = 1 is the equation defining the unit circle.
- in the chebyshev metric, points have length 1 if one of the coordinates has length 1 and the other coordinates have a length smaller (or equal to) 1. and these conditions also describe the square with sides x = ± 1 and y = ± 1.
Chebyshev distance can also be called chessboard distance if you want something more descriptive.
Is that so? Why would some spells specify geometry then? For example fireball says it is a 20 ft radius while Hallucinatory Terrain specifies that it affects a 150 ft cube which, under Chebyshev distance, would be the same as a sphere right? My understanding was that D&D 5e uses euclidean distance with a minimum threshold of a square that has to be covered to be counted.
D&D's targeting rules are quite strange, but yes, it's very explicit that Chebyshev is used in 5e by default, if playing on a grid. On page 192 of the 5.0e PHB:
To enter a square, you must have at least 1 square of movement left, even if the square is diagonally adjacent to the square you're in.
The DMG presents, on page 252, an optional variant of the optional grid rules, which is to treat it the same as Pathfinder 2e does (alternating 5 ft and 10 ft):
The Player's Handbook presents a simple method for counting movement and measuring range on a grid: count every square as 5 feet, even if you're moving diagonally. ... This optional rule provides more realism.
When measuring range or moving diagonally on a grid, the first diagonal square counts as 5 feet, but the second one counts as 10 feet. This pattern...continues when you're counting diagonally even if you move horizontally or vertically between different bits of diagonal movement.
As for the value of cube vs sphere in the context of Chebyshev ranges, there are two key differences.
First, cubes measure side length, spheres measure radius. A 10 ft cube covers 4 squares. A 10 ft sphere covers 16.
Second, and more importantly (since the above could easily be translated by using only cubes or only spheres throughout the system, with either half or double the numbers), cubes are cast from one side, whereas spheres are cast from the centre. If you're standing in the front line with enemies in front of you and allies behind, a cube cast with you as its origin point will hit either allies only or enemies only, but not both. A sphere cast with you at its origin point will affect both allies and enemies. Note that the rules for cube, on page 204 of the 5.0 PHB say "A cube's point of origin is not included in the cube's area of effect, unless you decide otherwise." So you could include yourself and your allies, or you could include enemies but not yourself, if you so desired. Or, less likely, you could include allies but not yourself, or enemies and yourself.
From memory, cube spells are mostly cast from a range of "self", which is where this becomes an important distinction. If a spell has a range of X feet and cube, then the main difference is just that its area is smaller but its reach is longer than a sphere with the same numbers.
That's really stupid but apparently I've been playing wrong this whole time.
Fireball says radius, but in a non-Euclidian geometry radius doesn't translate to a Euclidian sphere. Embrace the cube of constant radius!
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There's no grid in the sky, though
Fair point. I actually don't know what, if anything, the D&D (or Pathfinder) rules say on this matter. I've always just treated it as a natural 3D extension of the 2D grid rules. If they're three squares in one direction, same square in the other, and 10 feet up, I'd treat that as 15 feet away because of Chebyshev rules.
I’ve always just treated it as a natural 3D extension of the 2D grid rules
I believe that's how it's handled in D&D too, or at least how my table has always done it. I meant more as a practical matter, you're very unlikely to have a vertical wall grid and some kind of stand of the correct height for your minis, so you can't just count squares like you would for horizontal movement. That's when the Pythagorean Theorem comes up in my experience.
In PF1e you'd still alternate between Manhattan and Chebyshev. I used to know the rules to that so well I'd run it without the book for reference.
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Depends how tall they are.
Holy shit it does.
The halfling and dwarf are out of luck. Human stands a chance.
If the cleric is 30ft in the air, and the allies are 20ft away but on the ground, then the allies are probably 10ft tall
If the range of Bless is 30ft and the Cleric is 30ft in the air, then any non-zero horizontal distance would technically put them out of range. You don't need to calculate that they are 36.06ft away to know if they are out of range or not.
But do their feet need to be in range or just a single part of their body?
Depends on which part of them needs to be blessed?
The opposite of that happened to Achilles
Except the game uses Chebyshev distance, so as long as they're within 30 feet in the x, y, and z dimensions, they're within 30 feet.
Though for area damage spells, it's much, much more complicated. You don't just have to find the Euclidean distance from them to the center. You have to calculate how much of their square is within that distance.
Though for area damage spells, it’s much, much more complicated.
That's an optional variant rule described in Xanathar's Guide. The default rule for grids is simpler: just do Chebyshev.
Do some people actually playing RPG care that much about range ? Rather than some guesstimate ?
I actually find the Ryuytama range management pretty cool, where you simply say whether your character is at contact/short-range/long-range/away and that's it.
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Some do. I like positioning on a grid as a part of combat. It rewards tight play and understanding the mechanics. When I DM though, it depends on the playgroup. I think most people prefer guesstimating and just applying the rule of cool
A wild Ryutama reference is crazy. What a charming system.
It always comes up at some point. Most DMs will either handwave or give a generous approximation. Inexperienced DMs (or those that just run a tight ship) will actually calculate it.
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