Hypotenuse: Definition, Properties, History, and Applications

Key properties

The fundamental relation that determines the hypotenuse is the Pythagorean theorem: a² + b² = c², so the hypotenuse length c equals the square root of (a² + b²). For example, a triangle with legs 3 and 4 has a hypotenuse of 5 (a well-known Pythagorean triple). The hypotenuse is opposite the 90° angle; more on the right angle is available at right-angle basics.

Origin and historical notes

The term "hypotenuse" derives from Greek and was used in classical geometry to designate the side stretching under the right angle. The relationship between the square of the hypotenuse and the squares of the legs is attributed to the school of Pythagoras, although equivalent results appear in earlier Babylonian and Indian mathematics. For a short historical survey see historical context.

Uses and examples

The hypotenuse plays a central role in trigonometry: sine and cosine are often defined relative to the hypotenuse (e.g., sine = opposite/hypotenuse). This connection makes the hypotenuse essential in solving right-triangle problems in surveying, construction, navigation, computer graphics and physics. For applied methods and formulas consult trigonometry resources.

Related concepts and notable facts

Only right triangles have a hypotenuse; in non-right triangles no side is designated as such. A noteworthy geometric fact: the hypotenuse of a right triangle is the diameter of its circumcircle (Thales' theorem and its converse). More advanced interpretations view the Pythagorean relation as a statement about Euclidean distance in two dimensions and a template for generalizations to higher-dimensional distance formulas. For further reading and references see additional material.

• Typical notation: sides a, b (legs); c (hypotenuse).
• Common example: 3–4–5 triangle.
• Essential formula: c = sqrt(a² + b²).