In a

right 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 a ...

, a cathetus (originally from the

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 ...

word ; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to 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 ...

. It is occasionally called a "side about the right angle". The side opposite the right angle is the

hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...

. In the context of the hypotenuse, the catheti are sometimes referred to simply as "the other two sides". If the catheti of a right triangle have equal lengths, the triangle is

isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

. If they have different lengths, a distinction can be made between the minor (shorter) and major (longer) cathetus. The

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

of the lengths of the catheti defines the

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

tangent and cotangent of the

acute 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 ar ...

s in the triangle: the ratio

c_1/c_2

is the tangent of the acute angle adjacent to

c_2

and is also the cotangent of the acute angle adjacent to

c_1

. In a right triangle, the length of a cathetus is the

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

of the length of the adjacent segment cut by the

altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

to the hypotenuse and the length of the whole hypotenuse. By 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 opposit ...

, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse. The term leg, in addition to referring to a cathetus of a right triangle, is also used to refer to either of the equal sides of an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

or to either of the non-parallel sides of a

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...

. In architecture, the term cathetus has been used for the eye of the

volute A volute is a spiral, scroll-like ornament that forms the basis of the Ionic order, found in the capital of the Ionic column. It was later incorporated into Corinthian order and Composite column capitals. Four are normally to be found on an Ion ...

. It was so termed because its position is determined, in an Ionic (or voluted)

capital Capital may refer to: Common uses * Capital city, a municipality of primary status ** List of national capital cities * Capital letter, an upper-case letter Economics and social sciences * Capital (economics), the durable produced goods used fo ...

, by a line let down from the point in which the volute generates.

References

External links

* Bernhardsen, T. ''Geographic Information Systems: An Introduction'', 3rd ed. New York: Wiley, p. 271, 2002.
''Cathetus'' at Encyclopaedia of Mathematics
* {{MathWorld , urlname=Cathetus , title=Cathetus Parts of a triangle