A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1⁄4 turn or 90 degrees).

The side opposite to the right angle is called the hypotenuse (side ${\displaystyle c}$ in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side ${\displaystyle a}$ may be identified as the side adjacent to angle ${\displaystyle B}$ and opposite (or opposed to) angle ${\displaystyle A,}$ while side ${\displaystyle b}$ is the side adjacent to angle ${\displaystyle A}$ and opposite angle ${\displaystyle B.}$

Every right triangle is half of a rectangle which has been divided along its diagonal. When the rectangle is a square, its right-triangular half is isosceles, with two congruent sides and two congruent angles. 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. This is Thales' theorem.

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, ${\displaystyle a^{2}+b^{2}=c^{2}.}$ If the lengths of all three sides of a right triangle are integers, the triangle is called a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.

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 ${\displaystyle c}$ is the length of the hypotenuse (side opposite the right angle), and ${\displaystyle a}$ and ${\displaystyle b}$ are the lengths of the legs (remaining two sides). Pythagorean triples are integer values of ${\displaystyle a,b,c}$ satisfying this equation. 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."