Hypotenuse
TrigonometryThe hypotenuse is the longest side of a right triangle, located opposite the right angle.
Formula
c = \sqrt{a^2 + b^2}
Definition
The hypotenuse is the longest side of a right triangle. It is always the side directly across from the corner that makes the little square (the right angle).
Example
In a triangle with sides $3$, $4$, and $5$, the side of length $5$ is the hypotenuse because it sits across from the $90$-degree corner.
Key Insight
No matter how you flip or rotate a right triangle, you can always find the hypotenuse by looking for the side opposite the right-angle corner.
Definition
The hypotenuse is the side of a right triangle opposite the $90$-degree angle. It is always the longest side, and its length $c$ is found using the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the legs.
Example
A right triangle has legs of $5$ and $12$. The hypotenuse $= \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
Key Insight
The hypotenuse is the denominator in both the sine and cosine ratios: $\sin = \text{opposite}/\text{hypotenuse}$ and $\cos = \text{adjacent}/\text{hypotenuse}$. It anchors all basic trig ratios.
Definition
In a right triangle with legs $a$, $b$ and hypotenuse $c$, the relationship $c^2 = a^2 + b^2$ follows from the Euclidean metric. In the unit circle formulation, the hypotenuse of the reference right triangle is always $1$, making $\sin(t)$ and $\cos(t)$ the legs directly, which explains why $\sin^2(t) + \cos^2(t) = 1$.
Example
In the complex plane, the modulus $|z| = \sqrt{a^2 + b^2}$ for $z = a + bi$ is precisely the hypotenuse of the right triangle formed by the real and imaginary parts. The Pythagorean theorem generalizes to inner product spaces via $||v||^2 = \langle v, v \rangle$.
Key Insight
The hypotenuse is the geometric mean connection between altitude and segments of a right triangle (geometric mean altitude theorem), a result with deep implications in projective geometry.