Right Triangle

Right Triangle

A right triangle is a triangle where one of the three angles is a perpendicular angle. One only needs one of the non-perpendicular angles to define the shape of a right triangle, denoted above using the symbol . There are three sides of the right triangle relative to the angle : the adjacent, opposite, and hypotenuse sides. In math, angles usually are introduced using the unit degrees and then later the unit of radians.

Pythagorean Theorem

A well known property of the right triangle is Pythagorean’s Theorem which relates the squared-length of the adjacent, opposite, and hypotenuse sides together.

Pythagorean's Theorem

Special Right Triangles

There are two special right triangles which have simple, unique properties.

45 45 90 Triangle

Special Right Triangle 45 45 90

The 45° 45° 90° right triangle is a special triangle where the adjacent and opposite sides are equal in length. The most simple version of this triangle is where the adjacent and opposite sides are length 1 and then, using Pythagorean’s theorem, the hypotenuse is equal to the .

30 60 90 Triangle

Special Right Triangle 30 60 90

The 30° 60° 90° right triangle is a special triangle where the hypotenus is twice the length of the opposite side. If the opposite side is of length 1 and the hypotenuse is of length 2, then using Pythagorean’s theorem, then length of the adjacent side is equal to .

Trigonometric Functions

Right Triangle

There are three main trigonometric functions that relate the angle of a right triangle with the ratio of its sides. They symbol θ is used to denote the angle.

Cosine | Function

Given the angle of a right triangle as input, returns the ratio of the triangle's adjacent side over its hypotenuse.

Sine | Function

Given the angle of a right triangle as input, returns the ratio of the triangle's opposite side over its hypotenuse.

Tangent | Function

Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.

Further Reading

The right-triangle connects to many other concepts. It is the shape that makes up a 2D point in the Cartesian Coordinate System. It is the shape that is formed by a 2D vector and so on. Here are some links.

Cartesian Coordinate System | Concept

The Cartesian Coordinate System describes space of one, two, and three dimensions. Each point in space is represented by its distance relative to the origin of the system.

Unit Circle | Concept

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.

Trigonometric Identities | Concept

The trigonometric identites are a set of equations derived from the properties of the right triangle.