45-45-90 Triangle
A 45-45-90 triangle is a special right triangle with angles of $45$, $45$, and $90$ degrees, where the legs are equal and the hypotenuse is $\sqrt{2}$ times a leg.
Formula
\text{legs: } a, a; \quad \text{hypotenuse: } a\sqrt{2}
Definition
A $45$-$45$-$90$ triangle is an isosceles right triangle with two $45^\circ$ angles and one $90^\circ$ angle; both legs are equal length $a$, and by the Pythagorean theorem the hypotenuse $= a\sqrt{2}$, since $a^2 + a^2 = 2a^2 = (a\sqrt{2})^2$. Key trig values: $\sin(45^\circ) = \cos(45^\circ) = 1/\sqrt{2} = \sqrt{2}/2 \approx 0.707$. On the unit circle, it corresponds to the angle $\pi/4$, with terminal point $(\sqrt{2}/2, \sqrt{2}/2)$ lying on the line $y = x$, the axis of symmetry that reflects the unit circle to itself, explaining why $\sin(\pi/4) = \cos(\pi/4)$.
Example
If each leg is $5$ cm, the hypotenuse $= 5\sqrt{2} \approx 7.07$ cm, and you can check $5^2 + 5^2 = 50 = (5\sqrt{2})^2$. A square with diagonal $10$ has side $= 10/\sqrt{2} = 5\sqrt{2} \approx 7.07$, since the diagonal of any square creates two $45$-$45$-$90$ triangles. The eigenvalues of the reflection matrix across $y = x$ are $+1$ and $-1$, with eigenvectors along the $45^\circ$ and $135^\circ$ directions, encoding the triangle's geometry in this reflection symmetry.
Key Insight
This triangle is exactly half of a square cut along the diagonal, which is why both legs are equal, and $45^\circ$ is the only acute angle where sine equals cosine, making it a symmetric, special case on the unit circle. The side ratio $1:1:\sqrt{2}$ determines the lattice constant of the Eisenstein-Jacobi lattice, and in crystallography, the face-centered cubic structure is based on this ratio, connecting the special triangle to materials science.