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For example, if one of the legs of a right angle has a length of 3 and the other has a length of 4, then their squares add up to 25 = 9 + 16 = 3 × 3 + 4 × 4. Since 25 is the square of the hypotenuse, the length of the hypotenuse is the
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As a consequence of the
Pythagorean theorem, the hypotenuse is the longest side of any right triangle; that is, the hypotenuse is longer than either of the triangle's legs. For example, given the length of the legs
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761:= 12, then the sum of the legs squared is (5 × 5) + (12 × 12) = 169, the square of the hypotenuse. The length of the hypotenuse is thus the square root of 169, denoted
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of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. Mathematically, this can be written as
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Some languages have extended the definition to higher dimensions. For example, C++17 supports
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1035:{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \theta =a^{2}+b^{2}\implies c={\sqrt {a^{2}+b^{2}}}.}
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equals the hypotenuse squared. In mathematical notation, with the respective legs labelled
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1849:"hypotenuse | Origin and meaning of hypotenuse by Online Etymology Dictionary"
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The
Pythagorean theorem, and hence this length, can also be derived from the
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Longest side of a right-angled triangle, the side opposite of the right angle
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of the leg adjacent of the angle and the hypotenuse. For a right angle
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1137:{\displaystyle {\mbox{std::hypot}}(x,y,z)={\sqrt {x^{2}+y^{2}+z^{2}}}}
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1917:. Undergraduate Texts in Mathematics. New York: Springer. p. 133.
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Many computer languages support the ISO C standard function hypot(
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Some scientific calculators provide a function to convert from
1639:{\displaystyle \beta \ =\arccos \left({\frac {a}{c}}\right)\,}
1443:{\displaystyle \beta \ =\arcsin \left({\frac {b}{c}}\right)\,}
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being the feminine present active participle of the verb
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1770:. Vol. 27 (11th ed.). 1911. p. 258.
145:. Unsourced material may be challenged and removed.
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may be too technical for most readers to understand
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879:{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \theta }
1238:, one can obtain the value of two acute angles,
1913:Millian, Richard S.; Parker, George D. (1981).
1378:{\displaystyle {\frac {b}{c}}=\sin(\beta )\,}
1184:to handle an arbitrary number of arguments.
1144:; this gives the length of the diagonal of a
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1567:One may also obtain the value of the angle
606:, while the other two sides are called the
53:Learn how and when to remove these messages
1214:. The angle returned is normally given by
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239:A right-angled triangle and its hypotenuse
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273:of the hypotenuse can be found using the
223:Learn how and when to remove this message
205:Learn how and when to remove this message
103:Learn how and when to remove this message
87:, without removing the technical details.
742:{\displaystyle c={\sqrt {a^{2}+b^{2}}}.}
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1915:Geometry: A Metric Approach with Models
1873:"hypotenuse definition and word origin"
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1392:The trigonometric inverse function is:
354:of 25, that is, 5. In other words, if
587:A right triangle with the hypotenuse
524:"to stretch below, to subtend", from
85:make it understandable to non-experts
7:
143:adding citations to reliable sources
1478:is the angle opposite the cathetus
1283:Given the length of the hypotenuse
1206:above) at the same time when given
1177:{\displaystyle {\mbox{math.hypot}}}
1502:The adjacent angle of the catheti
342:is the length of another leg, and
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1740:Norm_(mathematics)#Euclidean_norm
686:{\displaystyle a^{2}+b^{2}=c^{2}}
346:is the length of the hypotenuse.
327:{\displaystyle a^{2}+b^{2}=c^{2}}
34:This article has multiple issues.
1784:Jr, Jesse Moland (August 2009).
1684:
119:
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2025:at Encyclopaedia of Mathematics
1900:(1520), fol. 221r (cited after
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42:or discuss these issues on the
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778:{\displaystyle {\sqrt {169}}}
1790:. Jesse Moland. p. 1.
1761:"Triangle (geometry)"
796:. In a right triangle, the
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1948:Linux Programmer's Manual
1923:10.1007/978-1-4684-0130-1
1537:{\displaystyle \alpha \,}
1280:, of the right triangle.
1252:{\displaystyle \alpha \,}
549:54d). The Greek term was
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1558:{\displaystyle \beta \,}
1471:{\displaystyle \beta \,}
1273:{\displaystyle \beta \,}
620:function implied by the
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1832:A Greek–English Lexicon
1767:Encyclopædia Britannica
1730:Special right triangles
1674:is the other cathetus.
1189:rectangular coordinates
569:, is French in origin (
1998:Python Language Manual
1896:Estienne de La Roche,
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2069:Pythagorean theorem
2059:Parts of a triangle
1994:"Python math.hypot"
1973:C++ Language Manual
1667:{\displaystyle a\,}
1516:{\displaystyle b\,}
1492:{\displaystyle b\,}
1318:{\displaystyle b\,}
1297:{\displaystyle c\,}
800:of an angle is the
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622:Pythagorean theorem
480:ἡ τὴν ὀρθὴν γωνίαν
399:{\displaystyle b=4}
373:{\displaystyle a=3}
275:Pythagorean theorem
2032:Weisstein, Eric W.
1877:Collins Dictionary
1853:www.etymonline.com
1692:Mathematics portal
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2053:Categories
2023:Hypotenuse
2016:References
1953:4 December
1944:"hypot(3)"
1883:2022-04-12
1858:2019-05-14
1735:Pythagoras
1170:math.hypot
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559:hypotēnūsa
555:Late Latin
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165:newspapers
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