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Burning hands states:

As you hold your hands with thumbs touching and fingers spread, a thin sheet of flames shoots forth from your outstretched fingertips.

However, it goes on to affect a cone with a 15 foot diameter base (which is certainly not a thin sheet).

Each creature in a 15-foot cone must make a Dexterity saving throw

What area can actually affect creatures, the thin sheet or the cone? Is it different for objects?

The fire ignites any flammable objects in the area that aren't being worn or carried.

For a use case: Does a tiny creature flying 10 feet away and 10 feet in the air get hit by the 5 foot tall caster's straight burning hands? Does the carpet below catch fire?

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