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How does the distance formula in three dimensions follow the Pythagorean theorem?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclidean geometry, the distance formula in three dimensions is a direct extension of the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.12

To find the distance between two points $(a, b, c)$ and $(d, e, f)$ in three-dimensional space, one essentially applies the Pythagorean theorem twice. First, the distance between the points in the $x y$ -plane is determined using the two-dimensional formula $\sqrt{(a - d)^{2} + (b - e)^{2}}$ . When this horizontal distance is treated as one leg of a new right triangle—with the vertical difference $(c - f)$ as the second leg—the hypotenuse represents the straight-line distance between the two points in 3D space.12

By squaring these components, the formula becomes:
$\sqrt{(a - d)^{2} + (b - e)^{2} + (c - f)^{2}}$
This calculation effectively generalizes the relationship between the sides of a right triangle to three-dimensional Euclidean space.12