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<math display=block>d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+\cdots+(p_i - q_i)^2+\cdots+(p_n - q_n)^2.</math>
Selain penerapannya dalam membandingkan jarak, jarak Euklides kuadrat merupakan alat penting di bidang [[statistika]]. Jarak tersebut dipakai dalam metode [[kuadrat terkecil]], sebuah metode statistik penyuaian (''method of fitting statistical'') yang mengestimasi data dengan meminimum rerata dari jarak kuadrat di antara nilai yang diamati dan nilai yang diestimasi,<ref>{{citation|title=Basic Statistics in Multivariate Analysis|series=Pocket Guide to Social Work Research Methods|first1=Karen A.|last1=Randolph|author1-link=Karen Randolph|first2=Laura L.|last2=Myers|publisher=Oxford University Press|year=2013|isbn=978-0-19-976404-4|page=116|url=https://books.google.com/books?id=WgSnudjEsrMC&pg=PA116}}</ref> dan dipakai sebagai bentuk [[Deret divergen|divergensi]] sederhana untuk membandingkan [[distribusi probabilitas]].<ref>{{citation
| last = Csiszár | first = I. | author-link = Imre Csiszár
| doi = 10.1214/aop/1176996454
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| year = 1975}}</ref> The addition of squared distances to each other, as is done in least squares fitting, corresponds to an operation on (unsquared) distances called [[Pythagorean addition]].<ref>{{citation |author=Moler, Cleve and Donald Morrison |title=Replacing Square Roots by Pythagorean Sums |journal=IBM Journal of Research and Development |volume=27 |issue=6 |pages=577–581 |year=1983 |url=http://www.research.ibm.com/journal/rd/276/ibmrd2706P.pdf |doi=10.1147/rd.276.0577 | citeseerx = 10.1.1.90.5651 }}</ref> Dalam [[analisis kluster]], jarak kuadrat dapat dipakai untuk memperkuat efek dari jarak yang lebih panjang.<ref name=spencer>{{citation|title=Essentials of Multivariate Data Analysis|first=Neil H.|last=Spencer|publisher=CRC Press|year=2013|isbn=978-1-4665-8479-2|contribution=5.4.5 Squared Euclidean Distances|page=95|contribution-url=https://books.google.com/books?id=EG3SBQAAQBAJ&pg=PA95}}</ref>
Jarak Euklides kuadrat tidak membentuk ruang metrik, sebab tidak memenuhi ketaksamaan segitiga.<ref>{{citation|last1=Mielke|first1=Paul W.|last2=Berry|first2=Kenneth J.|editor1-last=Brown|editor1-first=Timothy J.|editor2-last=Mielke|editor2-first=Paul W. Jr.|contribution=Euclidean distance based permutation methods in atmospheric science|doi=10.1007/978-1-4757-6581-6_2|pages=7–27|publisher=Springer|title=Statistical Mining and Data Visualization in Atmospheric Sciences|year=2000}}</ref> Akan tetapi, jaraknya bersifat mulus, yakni berupa [[fungsi cembung]] sempurna dari dua titik, tidak seperti jarak biasa yang tidak mulus (mendekati pasangan dari titik yang sama) dan cembung (tetapi bukan cembung sempurna). Karena itu, jarak Euklides kuadrat seringkali
Kumpulan dari semua jarak kuadrat di antara pasangan titik dari himpunan terhingga dapat disimpan dalam sebuah [[matriks jarak Euklides]], serta dipakai ke bentuk tersebut dalam geometri jarak.<ref>{{citation|title=Euclidean Distance Matrices and Their Applications in Rigidity Theory|first=Abdo Y.|last=Alfakih|publisher=Springer|year=2018|isbn=978-3-319-97846-8|page=51|url=https://books.google.com/books?id=woJyDwAAQBAJ&pg=PA51}}</ref>
== Perumuman ==
Dalam cabang matematika lebih lanjut, saat jarak Euklides dipandang sebagai [[ruang vektor]], jaraknya diiringi dengan [[Norma (matematika)|norma]] yang disebut sebagai [[Norma (matematika)#Norma Euklides|norma Euklides]], didefinisikan sebagai jarak dari masing-masing vektor yang berawal dari [[Titik asal (matematika)|titik asal]]. One of the important properties of this norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin.<ref>{{citation|title=Relativistic Celestial Mechanics of the Solar System|first1=Sergei|last1=Kopeikin|first2=Michael|last2=Efroimsky|first3=George|last3=Kaplan|publisher=John Wiley & Sons|year=2011|isbn=978-3-527-63457-6|page=106|url=https://books.google.com/books?id=uN5_DQWSR14C&pg=PA106}}</ref>
* [[Jarak Chebyshev]], yang menghitung jarak yang hanya dengan mengasumsi bahwa dimensi yang sangat signifikan adalah penting.
* [[Jarak Manhattan]], yang menghitung jarak hanya dengan mengikuti arah pada sumbu.
* [[Jarak Minkowski]], suatu perumuman yang menyatukan jarak Euklides, jarak Manhattan, dan jarak Chebyshev.
For points on surfaces in three dimensions, the Euclidean distance should be distinguished from the [[geodesic]] distance, the length of a shortest curve that belongs to the surface. In particular, for measuring great-circle distances on the earth or other spherical or near-spherical surfaces, distances that have been used include the [[haversine distance]] giving great-circle distances between two points on a sphere from their longitudes and latitudes, and [[Vincenty's formulae]] also known as "Vincent distance" for distance on a spheroid.<ref>{{citation|title=Computing in Geographic Information Systems|first=Narayan|last=Panigrahi|publisher=CRC Press|year=2014|isbn=978-1-4822-2314-9|contribution=12.2.4 Haversine Formula and 12.2.5 Vincenty's Formula|pages=212–214|url=https://books.google.com/books?id=kjj6AwAAQBAJ&pg=PA212}}</ref>
==
The [[Pythagorean theorem]] is also ancient, but it could only take its central role in the measurement of distances after the invention of [[Cartesian coordinates]] by [[René Descartes]] in 1637. The distance formula itself was first published in 1731 by [[Alexis Clairaut]].<ref>{{citation|last=Maor|first=Eli|author-link=Eli Maor|isbn=978-0-691-19688-6|pages=133–134|publisher=Princeton University Press|title=The Pythagorean Theorem: A 4,000-Year History|url=https://books.google.com/books?id=XuWZDwAAQBAJ&pg=PA133|year=2019}}</ref> Because of this formula, Euclidean distance is also sometimes called Pythagorean distance.<ref>{{citation|last1=Rankin|first1=William C.|last2=Markley|first2=Robert P.|last3=Evans|first3=Selby H.|date=March 1970|doi=10.3758/bf03210143|issue=2|journal=[[Perception & Psychophysics]]|pages=103–107|title=Pythagorean distance and the judged similarity of schematic stimuli|volume=7|s2cid=144797925|doi-access=free}}</ref> Although accurate measurements of long distances on the earth's surface, which are not Euclidean, had again been studied in many cultures since ancient times (see [[history of geodesy]]), the idea that Euclidean distance might not be the only way of measuring distances between points in mathematical spaces came even later, with the 19th-century formulation of [[non-Euclidean geometry]].<ref>{{citation|last=Milnor|first=John|author-link=John Milnor|doi=10.1090/S0273-0979-1982-14958-8|issue=1|journal=[[Bulletin of the American Mathematical Society]]|mr=634431|pages=9–24|title=Hyperbolic geometry: the first 150 years|volume=6|year=1982|doi-access=free}}</ref> The definition of the Euclidean norm and Euclidean distance for geometries of more than three dimensions also first appeared in the 19th century, in the work of [[Augustin-Louis Cauchy]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=[[Graduate Texts in Mathematics]]|first=John G.|last=Ratcliffe|edition=3rd|publisher=Springer|year=2019|isbn=978-3-030-31597-9|page=32|url=https://books.google.com/books?id=yMO4DwAAQBAJ&pg=PA32}}</ref>
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