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A druid in my game used the Entangle spell, and it was pretty easy to know how a 20' square rendered on a battle mat with a 5' spacing grid. Now he also knows Spike Growth...

The Spike Growth effect manifests in a circle with a 20' radius, which had us debating what happened to the outer corners of the square (this might be a visual exercise for some). Imagine a 20' square on a grid with 5' spacing. Now plant a circle with a 20' diameter over that square. The 4 5x5' corners of the square are barely touched by the circle so I figure that they aren't affected by the spell. Furthering my thought process is that if it did affect these grid squares, then there would be no mechanical difference between a 20' square and a 20' diameter circle, and there would be no need for wizards to word things inconsistently.

So, using a battle mat how does a DM interpret the "corners" of a circle. Are there mechanics for this that I am missing? Failing that, are there any good interpretations anyone has heard of or used that I could try?

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So, the best way to interpret this is to go back to the 3.5 version of D&D. In 4th Edition, there were no circular spell effects. Everything was a square to make things simple for the players and DM. Fifth Edition is going back to the earlier style of play where circles are more common, but on a grid, a circle isn't a circle at all (and they are not like the earlier poster submitted).

The "interpreted" circles are in the older Player's Handbooks and DM Guides, and they have been put up on the net in a number of places.

Here is one of the better guides that comes directly from an earlier stack exchange query (image below): What is a radius on a square grid?

And here is another PDF that one can cut for use on a battle map: http://www.enworld.org/forum/showthread.php?276089-Area-of-Effect-Templates

These exact templates are very likely to appear in the 5e Dungeon Master's Guide when it is out in November/December.

spell templates from earlier stack exchange answer

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    \$\begingroup\$ It's good to see these, but I find problems with some of them. The area of effect for the cone attacks aren't equal between the same sizes (15' cone, 6-squares when fired diagonally, 7-squares when shot in cardinal directions). My druid was adamant to show me some math for the circles and I think the AOE would extend to 8 more squares on the 30' circle (the ones on either side of the 2 most distant affected squares in each direction) as 69.5% of those squares are actually encompassed by the circle. \$\endgroup\$ – Besty Nov 8 '14 at 3:46
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    \$\begingroup\$ @Betsy those circles and squares seem made according to D&D 3.5e rules: start from a grid intersection and go from intersection to intersection, counting twice every other diagonal (so the distance on the major diagonal is 1/23/4/56 totalling 6 units over 4 diagonals. Of course the semplification isn't perfect. \$\endgroup\$ – Zachiel May 19 '15 at 21:45
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    \$\begingroup\$ Cones in 5e are not the same shape as cones in 3.5e. In 5e, "a cone's width at a given point along its length is equal to that point's distance from the point of origin." \$\endgroup\$ – PurpleVermont Aug 11 '15 at 3:57
  • \$\begingroup\$ Yeah, this might work for 3.5, but sure as hell seems factually wrong for the 5e rules. \$\endgroup\$ – J.E Jun 25 '18 at 6:44
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As the basis of a house rule, we found this math and diagram my druid-player drew up helpful to accurately figure out square coverage. (As I noted in a comment in Marty Walser's answer, some of the shapes in the the template he provided aren't mathematically correct, although neither is moving diagonally through a grid for the same distance). Click for a bigger version:

Area calculation for a circle on a grid

This obviously isn't a full answer, but currently there is no full answer provided by the game itself anyway. So this answer is just a tool you can use to solve the problem to your own satisfaction.

One way we've considered using this:

  • You could rule that any square with at least (e.g.) 50% coverage is affected.

    This gives different results from the templates based on city-block distance.The orange square (second from the right) has only 30.5% of its area outside of the circle, which one could use to conclude that the square is mostly inside of the circle, and thus conceivably affected by the spell — but using a city-block–distance template would count this square as safe from the spell.

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    \$\begingroup\$ The DMG p. 251 under the subsection "Areas of Effect" supports your comment above about a square being affected if at least 50% of it is covered, from the DMG: "If an area of effect is circular and covers at least half a square, it affects that square." \$\endgroup\$ – Sean Quinn Dec 22 '15 at 2:41
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Back in the 1st-edition/2nd-edition homew brew hybrid day I played a cleric from 1st up through pretty high levels, so I had lots of X-foot radius spell effects. There were not a lot of radii, though... mostly 5, 10 and 15 foot. So I printed these three circles (at 1" = 5 feet scale) onto clear transparencies (I used no grid since the grid of the battle mat showed through, although adding grid lines within the circles would be trivially easy to add), and cut them out, and they got placed on the battle mat. They worked pretty easily, and the only hassle was picking up figures and replacing them if they got cast onto an area where creatures and objects already were. Really, it worked out pretty decently, and answered very effectively whether or not something was in the circular area of effect. The printed disks allowed whatever was drawn on the mat underneath to show through the transparency, and had the added effect that one could easily draw or write things on temporarily them with the same water erasable pens that one typically uses on battle mats.

Edit: If a figurine or counter was on the lack circle printed on the transparency or within it was ruled to be within the area of effect, and was outside the area of effect otherwise.

One could also cut out the centers (i.e. make printed rings, rather than printed disks), which would offer the advantage of less need to pick up and set down figures when placing a circle, but the disadvantage of a more easily disturbed circle in the course of playing on the battle mat).

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    \$\begingroup\$ This doesn't appear to be answering the question of how the squares on the edge of the circle are supposed to be handled, particularly those that only barely overlap with the circle. \$\endgroup\$ – doppelgreener May 22 '15 at 1:35
  • \$\begingroup\$ @doppelgreener "Really, it worked out pretty decently, and answered very effectively whether or not something was in the circular area of effect." Seems to answer this question, albeit not by using the "treat squares as circles" method. I realize that my technique was unorthodox, but it unambiguously demarcates the circle boundaries. But perhaps I misunderstand your comment? \$\endgroup\$ – Lexible May 25 '15 at 18:10
  • \$\begingroup\$ There's a part missing from this answer: when you create your transparencies (I don't know if they're circles or grid-ified circles) or apply them, how do you handle squaures that the circle only partially overlaps, or just barely overlaps? I understand you have a useful aid, but that's the critical part of the inquiry. It sounds like you may have ignored that factor altogether and just paid attention to whether things on the map fell inside the circle, is that right? If so, expressing that specifically is missing. \$\endgroup\$ – doppelgreener May 25 '15 at 21:53
  • \$\begingroup\$ @doppelgreener Edited both to explicitly address within/without interpretation, and also to address whether gird-lines were printed on the transparent circles. Does that work better? \$\endgroup\$ – Lexible May 26 '15 at 16:08
  • \$\begingroup\$ @doppelgreener Thank you for the encouragement. :) \$\endgroup\$ – Lexible May 26 '15 at 16:12
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The PHB does not set out any mechanism for determining which squares are affected by a circular spell. However, based on vanilla movement rules, circles are square. Consider the grid movement box in the PHB (p 192):

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 rule for diagonal movement sacrifices realism for the sake of smooth play. The Dungeon Master's Guide provides guidance on using a more realistic approach.)

Under these simplified rules, it takes exactly as much movement to reach the "corner" of a circle as it does to reach a cardinal edge. The distinction in the text is probably there in anticipation of a different system in the DMG.

Although these rules make circles and squares identical they do not make cubes and spheres identical. Paraphrasing from PHB p 204, the point of origin of a cube is on one of its faces, the point of origin of a sphere is in the exact center of the sphere, and the point of origin of a cylinder is the center of the the top or bottom face.

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    \$\begingroup\$ Might be worth noting that using the movement rules for how to measure AoE coverage is not in the rules as written, but rather is a ruling suggestion. \$\endgroup\$ – SevenSidedDie Nov 9 '14 at 19:50
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There's no defined rule for this in 5e yet.

Two ways you could rule:

  1. The four furthest squares from the center of the circle are unaffected. This is how 3e did it.

  2. The four furthest squares from the center of the circle grant a save and/or bonus to avoid or mitigate the effects. This is a house rule.

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I cut rings out of ice cream pail tops and use those for consistent area for things like the fireball spell. As far as cones go, I made adjustable cones out of thick white board for things like a Cone of Cold or a Dragon breathing acid or fire. It works well and the players like it as they know where things are going in a combat situation.

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  • \$\begingroup\$ This doesn't appear to be answering the question of how the squares on the edge of the circle are meant to be handled, particularly those that only barely overlap with the circle. \$\endgroup\$ – doppelgreener May 22 '15 at 1:35
  • \$\begingroup\$ @doppelgreener, this actually works great. If you read the description of the spells, the spellcaster decides exactly of the location where the spell hits. So you give the circle / cone to the player and he / she places it exactly as he / she wants. Then you see what you hit. Way easier than counting squares or hexagons... \$\endgroup\$ – Alexis Wilke Oct 12 '15 at 7:08

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