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Burning hands says two things about the shape of the spell’s area of effect (PHB, p. 220):

  1. a thin sheet of flames shoots forth from your outstretched fingertips

  2. Each creature in a 15-foot cone [is affected by the spell]

These seem to conflict, as a thin sheet of flame and the definition of the cone AoE (PHB, p. 204) are different shapes. (And we know there’s no ignorable “flavour text” in spell descriptions, so that doesn’t help resolve it.)

What is the area of effect of burning hands?

(This came up in the context of answers to How to resolve Burning Hands with a Dying Ally.)

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The rules text say it is both. So it is both.

I can think of two reasonable ways to be both a cone and be a thin sheet.

First, a cone whose surface is a thin sheet of flames matches both descriptions. But here, it is hard to explain why something in the center of the cone would take damage (sure, radiant heat, but it just feels off).

Second, a cone in which a thin sheet of flame, perpendicular to the cone, travels from the origin out to the base of the cone matches both descriptions. In this case, damage over everyone in the cone makes perfect sense.

So, based off geometry, a thin disc of flame that starts at the origin and grows to 15' as it flies to the "base" of the cone matches all descriptions of the spell and effect without any contradictions.

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It's a cone

In two out of three points of the spell description, the term used is cone. The spell header has the following data.

Burning Hands; 1st-level evocation; Casting Time: 1 action; Range: Self (15-foot cone); Components: V, S; Duration: Instantaneous

As you note from the text, it mentions both "cone" and "thin sheet" in the full description.
Two votes for cone, one vote for sheet.

Two or three dimensions?

In the spell geometry discussion we find this under cone:

Cone
A cone extends in a direction you choose from its point of origin. A cone’s width at a given point along its length is equal to that point’s distance from the point of origin. A cone’s area of effect specifies its maximum length. A cone’s point of origin is not included in the cone’s area of effect, unless you decide otherwise (Basic Rules, p. 80)

AOE shapes

While that rules text may look like a two-dimensional shape on a grid, the illustration supports cone. (Three votes for cone). You could argue that it is not a three dimensional cone, but more like a truncated pyramid or a triangle when played on a grid. For theater of the mind play? It's a cone.

Plain English? It's a cone. Is the cone's whole volume necessarily full of flame? Unclear. See @Yakks' answer for why that may not be necessary.

DM call is needed here since the text isn't crystal clear

The rules text calls it a cone twice, a thin sheet once. A DM can rule reasonably that the three dimensional cone applies for a given situation like the one in the question you linked to. For most situations, the two dimensional and three dimensional cases are the same.

Area or Volume of effect?

Furthering the confusion, the text refers to the cone, and "area" versus "volume" for what is burned, or what is subjected to the magical effect. That may be done for ease of understanding by players and DM's who are not pedants, and who see a spell like burning hands (or fireball or cone of cold, as being in the class of "area of effect" spells rather than "single target spells."

As @DavidCoffron pointed out in a comment:

"area" in this case refers to the colloquial definition synonymous with region rather than the mathematical definition. The area of effect can have volume

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Functionally it is a 3D cone

While the description says it appears as though a thin sheet of flames shoots out (thin being subjective), for whatever reason you want to imagine it affects all creatures in a 15 foot 3D cone. It says that they all must save or take damage.

The two statements are not actually contradictions, and one actually describes the area hit and not only the shape of the visual effect. Potentially it's a thin sheet that moves rapidly around the cone, or rotates or passes across the area. For whatever reason you wish to flavour it as, the spell covers a 15 foot cone.

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It's a conic section. But that's being unnecessarily complex. It's easier to say that "When the game says 'cone' it doesn't necessarily mean 'cone' in the geometric sense."

Since everything is represented on a 2D map, and probably for historical reasons (Citation needed) effects that hit a triangle shape on the map are called "cones" by the game rules, probably because they were originally modeling stuff like cones of weapon fire. But as you have noticed, this being an actual 3D cone is inconsistent with the spell description.

Fortunately for all of us, however, it doesn't actually matter. In general, D&D doesn't track things on the Z-axis, so whether it's a "cone" in that sense or not is unlikely to matter -- unless you are dealing with an extremely low flying opponent, it's irrelevant. Also fortunately, it's easy to reconcile the "functional" aspect of a 3D cone with the fact that the description says it's a "sheet of flame" by assuming that the caster can "sweep" the sheet briefly over the entire area that would be covered by a 3D cone or that even though the flame itself doesn't cover the whole area, that it's HOT enough within the area to deal damage.

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  • The spell creates a thin sheet of flame.
  • The spell affects a conical volume.

These two assertions are both true. One way they can both be true is if the casting of the spell involves moving the sheet of flame. By rotating or sweeping the sheet of flame one could cause it to come in contact with everything in a conical volume.

(The reason why the sheet of flame cannot be moved across a volume of a different size or shape is, of course, one of the many obscure axioms of magic.)

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Its a cone formed by multiple sheets of flames coming from each finger. Probably expanding as all hot things do.

Think of each as if you have ten flamethrowers, one strapped to each finger in your hand. That would make a roughly conic shape overall, maybe a little flatter vertically, but not too much (hot air does go up).

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