A right triangle is a geometric figure with three sides, where two of those sides meet at a right angle. Right triangles are important in many areas of geometry and form the basis for trigonometry. The two sides of a right triangle that meet at the right angle are called the 'legs' and the third side is called the 'hypotenuse'.
The most important result regarding the right triangle is the Pythagorean Theorem, familiar from high school mathematics. It allows the determination of the length of any one side given the lengths of the other two sides. Similarly, the result that the three angles of any triangle add up to 180 degrees allow for the determination of the third angle given only the right angle and one of the other angles.
The relation between the sides and the angles is given by the trigonometric functions, sine, cosine, and tangent. Each of these functions is defined as the ratio between the lengths two of the sides of the triangle for a given angle. The sine is the ratio of the length of the leg opposing the angle to the length of the hypotenuse, and the cosine is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. It would be circular logic to use this to find side lengths were it not that both of these functions are continuous and that they have deep connections to other parts of mathematics. Thus, we can find the cosine and sine of an angle without constructing a triangle with that angle. With knowledge of these functions, the entire right triangle is determined by a single angle and a single side, or by any two sides.
Right triangles have many applications in mathematics and physics. The most important is probably coordinate systems. Both Cartesian and polar coordinates are based on right triangles with hypotenuses between a given point and the origin. Cartesian coordinates are the length of the two legs, polar coordinates are the length of the hypotenuse and the angle at the origin. They are connected through trigonometry and the Pythagorean theorem.
These coordinate systems are extended into three dimensions by adding another right triangle along the hypotenuse of the first. Combinations of descriptions of these triangles give the three common three-dimensional coordinate systems. If the lengths of the three independent legs are used as coordinates, three-dimensional Cartesian coordinates are obtained. Cylindrical coordinates are the the hypotenuse length and angle at the origin for the base triangle, and the length of the leg perpendicular to that triangle. Finally, if the angles of both triangles at the origin and the length of the second hypotenuse are used as coordinates then spherical coordinates are obtained. Please note that, although this is a valid set of spherical coordinates, used in geography, for example, mathematicians and physicists use a slightly different definition of one of the angles. See the spherical coordinates node for more details.
These coordinate systems can concievably be extended into further dimensions, but there is little practical use for any higher-dimensional coordinate system other than Cartesian coordinates, which can easily be generalised to an arbitrary number of dimensions.
All of these facts about right triangles necessarily assume a Euclidean manifold, and I am not qualified to discuss any other circumstances.
This writeup is copyright 2003 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .