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Hypotenuse
 
 
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, c1 and c2.]] A hypotenuse is the longest side of a right triangle, the side opposite the right angle
Right angle

In geometry and trigonometry, a right angle is an angle of 90 degree s, corresponding to a quarter turn . It can be defined; as the angle such that twice that angle amounts to a half turn, or 180?....
. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
, which states that the square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
 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 meters (when squared, 9 mē) and the other has a length of 4 m (when squared, 16 mē).






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, c1 and c2.]] A hypotenuse is the longest side of a right triangle, the side opposite the right angle
Right angle

In geometry and trigonometry, a right angle is an angle of 90 degree s, corresponding to a quarter turn . It can be defined; as the angle such that twice that angle amounts to a half turn, or 180?....
. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
, which states that the square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
 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 meters (when squared, 9 mē) and the other has a length of 4 m (when squared, 16 mē). Their squares add up to 25 mē. The length of the hypotenuse is the square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
 of this, or 5 m.

The word hypotenuse derives, according to some sources, from the Greek
Ancient Greek

Ancient Greek is the historical stage in the development of the Greek language spanning across the Archaic Greece , Classical Greece , and Hellenistic civilization periods of ancient Greece and the classical antiquity....
 ?p?te????sa (hypoteinousa), a combination of hypo- ("under") and teinein ("to stretch") . Others suggest the original meaning in Ancient Greek was for a thing which supports something in the manner of a prop or butress derived from a combination of hypo- ("under") and tenuse ("side").

Calculating the hypotenuse

Usually the length of the hypotenuse is calculated using the square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
 function in the obvious way. Setting x = c1 and y = c2 to avoid subscripts:

In mathematical notation;

Some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 the hypotenuse makes with the base line (c1 above) at the same time when given x and y.

See also

  • Cathetus
    Cathetus

    In a right triangle, the cathetus , most commonly known simply as a "leg" is either one of the two sides which are adjacent to the right angle in a right triangle....
  • Triangle geometry
  • Space diagonal
    Space diagonal

    In a cuboid or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner....
  • Nonhypotenuse number
    Nonhypotenuse number

    In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number cannot form the hypotenuse of a Pythagorean triple....
  • Taxicab geometry
    Taxicab geometry

    File:Manhattan distance.svgTaxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric space of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the differences of their coordinates....
  • Trigonometry
    Trigonometry

    Trigonometry is a branch of mathematics that deals with triangle s, particularly those plane triangles in which one angle has 90 degrees . Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships....