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What would be considered the threat range for attacks of opportunity when fighting an enemy flying above my PC?

Should I consider the adjacent squares threatened regardless of the height of the enemy's flight? Or is there a rule clarifying this in a more plausible way?

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Most distances in the game, reach included, are measured as radii from a point. While in typical play we use circles, radii actually define a 3D volume: a sphere. You determine whether or not you threaten something above you the same way you determine whether or not you threaten something on the same plane as you: by measuring your distance to it in 5-foot increments, or alternating 5-foot, 10-foot increments along diagonals (including diagonals that go up on the Z-axis).

For the most common reaches, 5 feet and 10 feet, however, the threatened volume is a cube, rather than a sphere. For 5 feet, at that distance the rounding of distances to 5-foot cubes means that all twenty-six cubes (eight around you, nine above, and nine below) are within reach, making a 3×3×3 cube around a character. For 10 feet, the special exception to the 5-foot, 10-foot alternating rule that allows a character with 10-foot reach to threaten at the corners (which would ordinarily be approximated as 15 feet away) also applies in 3D, making the threatened area a 5×5×5 cube.

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You threaten creatures within reach. No reason not to extend that to the Z-axis. For simplicity, I would assume a cubic shape such that each face is the 2D threatened area.

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