# Is “all targets must be within an X-unit diameter (circle)” equivalent to “all targets must be within X-units of each other”?

Several spells—or up-leveled versions of spells—have language such as this (charm person At Higher Levels from the Player's Handbook):

The creatures must be within 30 feet of each other when you target them.

I believe the above statement is equivalent to "The creatures must be within a 30-foot diameter (i.e. a 15-foot radius)."

When playing with such spells and using figurines and mats, I and others have sometimes started to measure distances between pairs of targets, rather than simply dropping a circle template of 30-foot diameter to see whether these can all fit inside.

Am I understanding correctly, and are the two phrases in my title exactly equivalent?

Note 1: I understand the difference between targeting an area of effect, like a circle or sphere, and targeting specific creatures within an area.

Note 2: Similar language appears in other editions, for example mass cure light wounds from Player's Handbook v3.5: "One creature/level, no two of which can be more than 30 ft. apart."

# No, "all targets within 30 feet of each other" is a more flexible criterion

We can show that the two criteria are not equivalent with a simple counterexample: suppose 3 targets are each 30 feet away from each other, such that they form an equilateral triangle with side length 30. The smallest circle that can enclose these 3 targets (known as the "circumcircle" of the triangle) has a diameter of of about 35 feet (see here; enter 30 for the edge length and press "Calculate").

However, going the other way, getting all targets inside a 35-foot circle is also not sufficient to establish that all targets are within 30 feet of each other. If two targets are at the edges exactly opposite each other, they will be 35 feet apart.

So, "all targets within X of each other" is not equivalent to "all targets within a circle of diameter X", and furthermore, there is no other diameter that you can substitute for X to make them equivalent.

In my answer, I have treated each target as a mathematical point with zero volume. It is fairly straightforward to extend my answer the targets with non-zero volumes, but it makes the math a bit more complicated, so I've opted to keep the idealized representation or targets as points for the sake of simplicity.

However, I should also note that my answer assumes that you are using Euclidean geometry. The Player's Handbook includes variant rules for playing on a grid, using quite non-Euclidean rules for calculating diagonal distances. As Mindwin's answer notes, these distance rules allow you to simplify the criterion of "all targets within 30 feet of each other". However, if you play on a grid with the variant diagonal rules from the Dungeon Master's Guide (putting diagonal distance at roughly 1.5x cardinal movement), or if you play using hexes instead of a square grid, this simplification no longer holds true.

### No. But you can swap the definition with a square if you are using the grid movement rules.

As Ryan proved in his answer, given three targets they have to be in a triangle with 30 feet of edge. But in the game this triangle cannot be drawn. The diagonals count as five foot of distance the same as the straight lines, so the best representation one can make is actually a 7 x 7 spaces square.

The distance to one's own square is 0 feet. The distance to the adjacent square is 5 feet.

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It is easy to prove that all squares inside this grid are within 30 feet of each other. Any square outside this will be more than 30 feet away from the opposite corner / edge.

If each diagonal costs 1.5x the movement (alternate DMG rules) then it becomes more like a circle:

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║xx║  ║xx║
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The cells marked with XX are actually 35 feet of each other. Four of them in opposed sides must be ignored. Thanks Ryan

• I would note that using a grid is presented in the PHB as a variant rule, so I believe Euclidean geometry is the "default". I believe your answer is correct for the grid rules presented in the PHB. There is also the additional variant in the DMG that makes diagonal distance effectively equal to 1.5x cardinal distance, which I think would complicate matters. Regardless, you make a good point that I should note that my answer is assuming Euclidean geometry. – Ryan Thompson Dec 3 '18 at 17:00
• @ryanthompson I feel like they slapped the 'variant' rule just to reinforce the notion that every rule is optional. When your own game supplements, official gameplay league and adventures all use these very 'variant' rules, they become, as Syndrome said, not a variant. However, I must add that remark. Thanks. – Mindwin Dec 3 '18 at 17:29
• I don't think that circle-ish shape quite works for the variant diagonal. Some spaces are 35 feet apart: a combination of 2 diagonal moves (15 feet) plus 4 cardinal moves (20 feet). My guess is that with the variant diagonal rules, the answer is probably that there is no shape for which "all targets within 30 feet of each other" is equivalent to "all targets inside this shape", but that seems quite hard to prove or disprove. – Ryan Thompson Dec 3 '18 at 19:22
• Yes, there are four cells that don't conform. I'll put an X on them. – Mindwin Dec 4 '18 at 11:26
• That helps a bit, but there are still cases it fails to cover. It's possible to find a set of 3 squares all 30 feet from each other measuring by the variant diagonal rules that cannot be covered by this shape - essentially the same logic as my answer for Euclidean geometry. So I'm led back to my unproven conjecture that the correct shape simply does not exist in this case. – Ryan Thompson Dec 4 '18 at 16:34

In the English language, those two phrase are not the same. Mathematically too.

Take for instance the Reuleaux triangle. It would allow all targets to be within 30 ft of each other, but it does not have a uniform 30ft radius. So in terms of pure topography, there can be differences.

And for the language, "creatures must be within 30 feet of each other when you target them", I would also take the more general meaning that targets do not all have to be in the same 30ft space, but only that each target is within 30ft of another target. So you could have multiple targets in a row each spaced 30ft apart. Or four targets on the corners of a 30ft square. Mathematically, the target directly diagonal is more than 30ft, but each adjacent target is only 30ft.

There is also a piece missing, where it could be that the import factor is "when you target them". Perhaps those spells you are looking at are not instantaneous and therefore the targets can move after targeting and they only need to be in proximity at the time the spell is cast. That would also make the meanings different.

• Not quite: there are points within the circle circumscribing a Reuleaux Triangle that are (a) within X-units of one another, and (b) not within the Reuleaux Triangle.... oh wait, I see we agree with the uniform radius. I have added the word "circle" to my title to clarify. – Lexible Dec 2 '18 at 22:03
• I'm not so sure this is also true, if you use the manhattan distance as d&d 5 does. At least for 3 targets I can prove it actually works – fabian Dec 2 '18 at 22:05
• This is false for RAW and RAI: "I would also take the more general meaning that targets do not all have to be in the same 30ft space, but only that each target is within 30ft of another target." – Lexible Dec 2 '18 at 22:05
• There is no other "general meaning" of the phrase. They must be within 30 feet of each other - a row of then separated by even 20 feet puts nonadjacent creatures up to 40 feet apart, clearly breaking the condition. – Nij Dec 2 '18 at 22:56
• MivaScott the phrasing "The creatures must be within 30 feet of each other when you target them" ≠ "The creatures must be within 30 feet of at least one other targeted creature when you target them. "Each other" refers to each member of a group, not to some members of the group in common English usege. – Lexible Dec 2 '18 at 23:37