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How is Euclidean distance calculated between two points on a coordinate plane?

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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 geometry, the distance between two points on a coordinate plane is calculated using a formula derived from the Pythagorean theorem.12

For any two points defined by their coordinates $(a, b)$ and $(c, d)$ on a Cartesian coordinate system, the Euclidean distance is given by the formula:

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

This calculation represents the length of the straight line segment connecting the two points. By forming a right triangle where the distance between the points serves as the hypotenuse, the formula effectively sums the squares of the horizontal and vertical differences between the points and takes the square root of that sum. This method is fundamental to analytic geometry and can be extended to three-dimensional space by including a third coordinate for each point.123