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How is distance calculated between two points in three-dimensional Euclidean space?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In Euclidean space, the distance between two points is defined as the length of the straight line segment connecting them. This calculation is derived from the Pythagorean theorem, which relates the sides of a right triangle to its hypotenuse.12

For two points in three-dimensional space defined by their rectangular coordinates, $(a, b, c)$ and $(d, e, f)$ , the distance is calculated using the following formula:

$\sqrt{(a - d)^{2} + (b - e)^{2} + (c - f)^{2}}$

This formula effectively extends the two-dimensional distance calculation into a third dimension by accounting for the difference in coordinates along the $x$ , $y$ , and $z$ axes. While this specific expression applies to rectangular coordinates, the underlying principle of Euclidean geometry can also be adapted for other systems, such as spherical coordinates.12