In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a hypotenuse is the longest side of a
right-angled triangle, the side opposite the
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
. The
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
of the hypotenuse can be found using the
Pythagorean theorem, which states that the
square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse is the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of 25, that is, 5.
Etymology
The word ''hypotenuse'' is derived from
Greek (sc. or ), meaning "
idesubtending the right angle" (
Apollodorus
Apollodorus (Greek: Ἀπολλόδωρος ''Apollodoros'') was a popular name in ancient Greece. It is the masculine gender of a noun compounded from Apollo, the deity, and doron, "gift"; that is, "Gift of Apollo." It may refer to:
:''Note: A f ...
), ''hupoteinousa'' being the feminine present active participle of the verb ''hupo-teinō'' "to stretch below, to subtend", from ''teinō'' "to stretch, extend". The nominalised participle, , was used for the hypotenuse of a triangle in the 4th century BCE (attested in
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, ''
Timaeus'' 54d). The Greek term was
loaned into
Late Latin
Late Latin ( la, Latinitas serior) is the scholarly name for the form of Literary Latin of late antiquity.Roberts (1996), p. 537. English dictionary definitions of Late Latin date this period from the , and continuing into the 7th century in t ...
, as ''hypotēnūsa''. The spelling in ''-e'', as ''hypotenuse'', is French in origin (
Estienne de La Roche 1520).
Calculating the hypotenuse
The length of the hypotenuse can be calculated using the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
function implied by the
Pythagorean theorem. Using the common notation that the length of the two legs of the triangle (the sides perpendicular to each other) are ''a'' and ''b'' and that of the hypotenuse is ''c'', we have
:
The Pythagorean theorem, and hence this length, can also be derived from the
law of cosines by observing that the angle opposite the hypotenuse is 90° and noting that its cosine is 0:
:
Many computer languages support the ISO C standard function hypot(''x'',''y''), which returns the value above.
The function is designed not to fail where the straightforward calculation might overflow or underflow and can be slightly more accurate and sometimes significantly slower.
Some scientific calculators provide a function to convert from
rectangular coordinates to
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
. This gives both the length of the hypotenuse and the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
the hypotenuse makes with the base line (''c
1'' above) at the same time when given ''x'' and ''y''. The angle returned is normally given by
atan2
In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive (''y'',''x'').
Trigonometric ratios
By means of
trigonometric ratios, one can obtain the value of two acute angles,
and
, of the right triangle.
Given the length of the hypotenuse
and of a cathetus
, the ratio is:
:::
The trigonometric inverse function is:
:::
in which
is the angle opposite the cathetus
.
The adjacent angle of the catheti
is
= 90° –
One may also obtain the value of the angle
by the equation:
:::
in which
is the other cathetus.
See also
*
Cathetus
*
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
*
Space diagonal
*
Nonhypotenuse number
*
Taxicab geometry
*
Trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
*
Special right triangles
*
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His poli ...
Notes
References
''Hypotenuse'' at Encyclopaedia of Mathematics*
{{wiktionary, hypotenuse
Parts of a triangle
Trigonometry
Pythagorean theorem