Right triangle

The sides adjacent to the right angle are called legs (or catheti, singular: cathetus).

may be identified as the side adjacent to angle

Every right triangle is half of a rectangle which has been divided along its diagonal.

When the rectangle is not a square, its right-triangular half is scalene.

Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at the apex and the hypotenuse as the base; conversely, the circumcircle of any right triangle has the hypotenuse as its diameter.

The legs and hypotenuse of a right triangle satisfy the Pythagorean theorem: the sum of the areas of the squares on two legs is the area of the square on the hypotenuse,

The relations between the sides and angles of a right triangle provides one way of defining and understanding trigonometry, the study of the metrical relationships between lengths and angles.

The three sides of a right triangle are related by the Pythagorean theorem, which in modern algebraic notation can be written where

is the length of the hypotenuse (side opposite the right angle), and

are the lengths of the legs (remaining two sides).

Pythagorean triples are integer values of

This theorem was proven in antiquity, and is proposition I.47 in Euclid's Elements: "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."

As with any triangle, the area is equal to one half the base multiplied by the corresponding height.

and the area is given by This formula only applies to right triangles.

[1] If an altitude is drawn from the vertex, with the right angle to the hypotenuse, then the triangle is divided into two smaller triangles; these are both similar to the original, and therefore similar to each other.

[3] Thus Moreover, the altitude to the hypotenuse is related to the legs of the right triangle by[4][5] For solutions of this equation in integer values of

The radius of the incircle of a right triangle with legs

is The radius of the circumcircle is half the length of the hypotenuse, Thus the sum of the circumradius and the inradius is half the sum of the legs:[6] One of the legs can be expressed in terms of the inradius and the other leg as A triangle

opposite the longest side, circumradius

is a right triangle if and only if any one of the statements in the following six categories is true.

The trigonometric functions for acute angles can be defined as ratios of the sides of a right triangle.

These include the 30-60-90 triangle which can be used to evaluate the trigonometric functions for any multiple of

be the harmonic mean, the geometric mean, and the arithmetic mean of two positive numbers

The converse states that the hypotenuse of a right triangle is the diameter of its circumcircle.

As a corollary, the circumcircle has its center at the midpoint of the diameter, so the median through the right-angled vertex is a radius, and the circumradius is half the length of the hypotenuse.

from the legs satisfy[6]: p.136, #3110 In a right triangle, the Euler line contains the median on the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex.

This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse.

trisect the hypotenuse into segments of length

be the sides of the two inscribed squares in a right triangle with hypotenuse

are related by a similar formula: The perimeter of a right triangle equals the sum of the radii of the incircle and the three excircles: