So I'm in a campaign and I'm getting stumped on how AoEs work on a grid. So we are playing in Roll20 and I'm asking the DM how they work. It seems he's trying to turn circles into squares, which would be fine, but there are some confusions. For example, he said moonbeam is essentially a 10 foot square on a grid. 5-foot radius equals 10-foot cube would be fine if every AoE has a point of origin in space. The problem is you have spells like ice knife where the point of origin is a creature. 

A solution I have to simplify this is the Roll20 site has a measuring tool. You can measure from one square to another. If you have an AoE radius, you can pick a center square and measure out to any creature that may be within. If the distance is less the the radius, they're affected. If not, they're not. The thing is it measures lines at angles as equivalent to vertical and horizontal ones, which means 5-foot radius would always be like a 15-foot cube. It's not a perfect solution but it's simpler and more consistent than the former. 

Cones on the other hand confound me. Say I use burning hands. I measure 15 feet out and 7.5 feet in both directions for the triangle base? Then I subtract one square as I go back? I haven't worked out a good way to figure out this AoE. 

I don't think the rules are very clear on how AoEs can work on a grid. If anyone can tell me a comprehensive and simple way to implement AoE rules for A grid, it would be appreciated. I can pitch it to my DM as an alternative to the idea he has, which I don't even understand myself. 

Any chance there's something similar for square grids?

It is nowhere near as elegant. Which is why hexes are better.

This is how we do it at my table. Note that we don't use Roll20, so I don't know how this is implemented there, sorry :(

If the point of origin is a creature, that creature occupies the entire square, and the range is calculated from the edge of its tile outwards. Here are some examples:

- Spirit Guardians (15-ft radius):  

- Burning Hands (15-ft cone): 

- Thunderwave (15-foot cube):    -  - 

- Earth tremor (10 ft):  

- Ice Knife (taking into acount the last erratum, 5 ft): 

If the point of origin is a point in space, then the drawing from Wysperra is how we do it. So, for example:

- Moonbean (5-ft radius): 

- Web (20-ft cube): 

About the number of affected targets, we follow the rules in XGtE (p. 86, Areas of Effect on a Grid):

If any part of a square is under the template, that square is included in the area of effect. If a creature's miniature is in an affected square, that creature is in the area. Being adjacent to the edge of the template isn't enough for a square to be included in the area of effect; the square must be entirely or partly covered by the template.

EDIT: added more examples.

The DMG provides some guidance for this on Page 251 in the sections called "Areas of Effect" and "Cover" (and some diagrams on nearby pages).  Perhaps not intuitively, the rule from the DMG says that when using a grid and miniatures you actually choose an intersection of grid squares as your point of origin:

AREAS OF EFFECT

The area of effect of a spell, monster ability, or other feature must be translated onto squares or hexes to determine which potential targets are in the area and which aren't.

Choose an intersection of squares or hexes as the point of origin of an area of effect, then follow its rules as normal. If an area of effect is circular and covers at least half a square, it affects that square.

There aren't any exceptions given for when a creature is the point of origin, so you end up with the point of origin on one of the corners of the square that the creature occupies.  This creates seemingly asymmetrical AoE shapes with respect to the creature's location within the square, but those are the rules as written.

So, for Moonbeam, your DM has it correct.  

In the case of Ice Knife, I see two options:

First, you can use the above rule to create an AoE with a radius of 5 feet, as indicated by the "Range/Area" parameter on the D&D Beyond spells database.  In cases where the point of origin is a creature, this creates a situation where a creature on one side will be affected but a nearby creature on the other side wouldn't be.  This sort of flies in the face of the text of the spell which reads "each creature within 5 feet . . .".

The second way is to consider this notation in the D&D Beyond spells database to be an error, since that notation doesn't exist in the hardcopy of the book anyway, and just stick with the text of the spell, which is that this AoE affects "each creature within 5 feet . . .".  However, in terms of trying to place an AoE like this onto a grid, you would have to break the rule for doing so that I've quoted above, or you would have to significantly enlarge the radius which could have other unintended consequences.

For a spell such as Burning Hands, you are definitely meant to choose a corner of a square as the origin point.  Because of the rule for cones that says that the point of origin is not included in the AoE, the spellcaster of Burning Hands is explicitly unaffected -- so, you could choose one of the back corners of your square if it's advantageous to do so.  The Xanathar's Guide to Everything book has two options for positioning shapes like this onto the grid once the point of origin is chosen -- the Template Method and the Token Method.  My guess is that the Template Method should work pretty well when using a tool such as Roll20 as you could then theoretically position the shape at any angle instead of getting locked into only vertical or diagonal options.  Note that the example given in Xanathar's for using the Template Method for a Cone clearly shows the point of origin at a grid intersection.

Also note that the rules given for the Template Method might make shapes such as Cones a lot more powerful than intended if they are angled just right given that the rule says:

". . .  take note of which squares it covers or partially covers. If any part of a square is under the template, that square is included in the area of effect. If a creature’s miniature is in an affected square, that creature is in the area."

This rule is in conflict with the original DMG rule for circular AoEs as quoted above, so a DM would have to make a ruling on that.

So, this is what the rules actually say on the matter.  If you are using an online tool such as Roll20 you might be further constrained by the capabilities of that tool.

@alexdohm9191 If you or your DM are interested in the Dev's opinion on interactions between ranges, grids, and points of origin, you might consider the following tweets:

@enormousturtle Do spells with a range of "Self" have a target? Is the caster the target? Trying to figure out Find Steed's spell sharing.
@JeremyECrawford A range of self means the caster is the target, as in shield, or the point of origin, as in thunderwave (PH, 202).

@samiam8910 How do spells with range Self (X-foot radius) work for bigger creatures? Is the radius from center or creature?
@JeremyECrawford When you create an area of effect with a range of self, your space is the point of origin, whatever your size.
@samiam8910 So an Ancient Dragon with Destructive Wave has the potential to wreck more than a Medium cleric doing the same? (Awesome)
@Jeremy Crawford That's correct.

@ThinkingDM Visualizing one of @JeremyECrawford's reasons why Centaur and Minotaur should not be Large PCs: a 5' aura is 50% bigger and a 10' aura is 33% bigger.

@JeremyECrawford That's exactly right.
@ThinkingDM Is it safe to assume we shouldn’t expect Large PC races?
@JeremyECrawford Yep.

@mrlong78 Thunderwave Spell:Where is the cube?  Is the caster at the center of the cube, or on one side of the cube? Caster's choice?
@JeremyECrawford The point of origin of a cubic area of effect, including thunderwave's, is on a face of the cube (PH, pg 204), not inside it.

@RaywkLam Which is correct representing the area of paladin aura (10 ft)?
@JeremyECrawford Is each square 5 ft., and which aura are you talking about?
@RaywkLam Yes, each square is 5 ft. I meant Aura of Protection and Aura of Courage. Thanks.
@JeremyECrawford Aura of Protection/Courage extends 10 ft. all around. (None are correct, unless your circles are polygons.)

@JeremyECrawford A note about D&D spells with a range of "Self (XYZ)": the parenthetical—which says "5-foot radius," "15-foot cone," or something else—means you are the spell's point of origin, but you aren't necessarily its target. You're creating an effect that originates in your space.

Some of the tweets are related to the following rules from the PHB:

Variant: Playing on a Grid (p. 192)

Ranges. To determine the range on a grid between two things—whether creatures or objects—start counting squares from a square adjacent to one of them and stop counting in the space of the other one. Count by the shortest route.

Range (Player's Handbook p.202, emphasis mine):

Spells that create cones or lines of effect that originate from you also have a range of self, indicating that the origin point of the spell's effect must be you.

Point of origin (Player's Handbook p. 204, emphasis mine):

Typically, a point of origin is a point in space, but some spells have an area whose origin is a creature or an object.

Note that this last rule does not state that some spells have points of origin 'in' or 'on' a creature or object. Instead, it specifies that the creatures or objects themselves are the origins of the spells. In these cases, the entire creature or object follows the rules regarding points of origin.

Quote from alexdohm9191 >>

So we are playing in Roll20

Roll20 has a way to handle AoEs of all shapes built into it already. You don't need to "figure out" how it would work -- you just pick 'Cone' from the Shapes menu instead of 'Line' in the measuring tool and measure out to 15 feet, and any token touched by the displayed AoE gets hit by your burning hands

So if I understand correctly, you're suggesting that if the point of origin is a point in space (such as moonbeam) that x-foot radius is resection 2x-foot cube (for moonbeam, 5-foot radius means 10-foot cube). But in cases where the point of origin is a creature (such as ice knife) it's effectively a 15-foot cube because it extended outward 5 feet from a square rather than a point in space?

Quote from AntonSirius >>

Quote from alexdohm9191 >>

So we are playing in Roll20

 

Roll20 has a way to handle AoEs of all shapes built into it already. You don't need to "figure out" how it would work -- you just pick 'Cone' from the Shapes menu instead of 'Line' in the measuring tool and measure out to 15 feet, and any token touched by the displayed AoE gets hit by your burning hands

 Is that available for players? Or the DM?

Quote from alexdohm9191 >>

Quote from AntonSirius >>

Quote from alexdohm9191 >>

So we are playing in Roll20

 

Roll20 has a way to handle AoEs of all shapes built into it already. You don't need to "figure out" how it would work -- you just pick 'Cone' from the Shapes menu instead of 'Line' in the measuring tool and measure out to 15 feet, and any token touched by the displayed AoE gets hit by your burning hands

 Is that available for players? Or the DM?

It should be available for everyone, unless there's some way for a DM to switch it off I'm not aware of

Do you see that Shapes menu on the measuring tool?

Quote from AntonSirius >>

Quote from alexdohm9191 >>

Quote from AntonSirius >>

Quote from alexdohm9191 >>

So we are playing in Roll20

 

Roll20 has a way to handle AoEs of all shapes built into it already. You don't need to "figure out" how it would work -- you just pick 'Cone' from the Shapes menu instead of 'Line' in the measuring tool and measure out to 15 feet, and any token touched by the displayed AoE gets hit by your burning hands

 Is that available for players? Or the DM?

 

It should be available for everyone, unless there's some way for a DM to switch it off I'm not aware of

Do you see that Shapes menu on the measuring tool?

 I knew I could measure lines, I didn't know I could measure shapes. I'll have to look through that when I get the chance. 

Quote from alexdohm9191 >>

So if I understand correctly, you're suggesting that if the point of origin is a point in space (such as moonbeam) that x-foot radius is resection 2x-foot cube (for moonbeam, 5-foot radius means 10-foot cube). But in cases where the point of origin is a creature (such as ice knife) it's effectively a 15-foot cube because it extended outward 5 feet from a square rather than a point in space?

 I’ve always operated that when you choose a “point” and you’re playing on a grid, the point is always at an intersection of 2 grid lines. So, yes, a 5’ radius means basically a 10’ square. Yes, this makes circles into squares. But I find it much easier than trying to decide what to do if a given square is only fractionally covered by the radius. So circles are squares and spheres are cubes. 

But even then, cones are just plain wonky. For them, we just kind do the best we can do.

Quote from AntonSirius >>

It should be available for everyone, unless there's some way for a DM to switch it off I'm not aware of

Do you see that Shapes menu on the measuring tool?

 I found it. I can do all the shapes. I'll let the DM know about this feature. Should simplify it. Thanks.