thumb|right|A right-angled triangle and its hypotenuse

In geometry, a hypotenuse is the side of a right triangle that is opposite to the right angle. It is always the longest side of the triangle. The other two sides of a right triangle are called legs or catheti.

The length 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. As an algebraic formula, this can be written as <math>a^2 + b^2 = c^2</math>, where is the length of one leg, is the length of the other leg, and is the length of the hypotenuse. For example, if the two legs of a right triangle have lengths 3 and 4, respectively, then the hypotenuse has length 5, because .

It was loaned into Latin as hypotenusa and later into French as hypoténuse. It first appeared in English in the 1570s. For example, if the two legs of a right triangle have lengths 3 and 4, respectively, then the hypotenuse has length 5, because . When is radians or 90°, then and the formula reduces to the usual Pythagorean theorem.

Sine and cosine

thumb|upright=1|For the angle , the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse

The sine and cosine functions (sin and cos) describe the relationship of the hypotenuse to the lengths and angles of the other two sides. These, along with tangent (tan), are the most common trigonometric functions. The function is designed not to fail where the straightforward calculation might overflow or underflow. It can often be more accurate and slower than the straightforward calculation. Python 3.8 includes a version of <code>math.hypot</code> which can handle an arbitrary number of arguments.

See also

  • Cathetus
  • Triangle
  • Space diagonal
  • Nonhypotenuse number
  • Taxicab geometry
  • Trigonometry
  • Special right triangles
  • Pythagoras
  • Euclidean norm

References

  • Hypotenuse at Encyclopaedia of Mathematics