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 A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...

, the side opposite the

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

. 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 Interna ...

of the hypotenuse can be found using the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, which states that the

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

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 . E ...

of 25, that is, 5.

Etymology

The word ''hypotenuse'' is derived from

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

(sc. or ), meaning " idesubtending the right angle" (

Apollodorus Apollodorus (Ancient Greek, 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: ...

), ''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 Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...

'' 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

function implied by the

. 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 :

c = \sqrt  .

The Pythagorean theorem, and hence this length, can also be derived from the

law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

by observing that the angle opposite the hypotenuse is 90° and noting that its cosine is 0: :

c^2 = a^2 + b^2 - 2ab\cos90^\circ = a^2 + b^2 \therefore c = \sqrt.

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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

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 the or ...

. This gives both the length of the hypotenuse and the

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

the hypotenuse makes with the base line (''c₁'' 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,

\alpha\,

and

\beta\,

, of the right triangle. Given the length of the hypotenuse

c\,

and of a cathetus

b\,

, the ratio is: Euklidova veta

:::

\frac = \sin (\beta)\,

The trigonometric inverse function is: :::

\beta\ = \arcsin\left(\frac  \right)\,

in which

\beta\,

is the angle opposite the cathetus

b\,

. The adjacent angle of the catheti

b\,

\alpha\,

= 90° –

\beta\,

One may also obtain the value of the angle

\beta\,

by the equation: :::

\beta\ = \arccos\left(\frac  \right)\,

in which

a\,

is the other cathetus.

Notes

References

''Hypotenuse'' at Encyclopaedia of Mathematics
* {{wiktionary, hypotenuse Parts of a triangle Trigonometry Pythagorean theorem

Etymology

Calculating the hypotenuse

Trigonometric ratios

See also

Notes

References