45-45-90 Triangle
TrigonometryA 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 a right triangle with two $45^\circ$ angles and one $90^\circ$ angle. Both legs are the same length, and the hypotenuse is about $1.41$ times longer than each leg.
Example
If each leg is $5$ cm, the hypotenuse $= 5\sqrt{2} \approx 7.07$ cm. You can check: $5^2 + 5^2 = 50 = (5\sqrt{2})^2$.
Key Insight
This triangle is exactly half of a square cut along the diagonal. That is why both legs are equal.
Definition
In a $45$-$45$-$90$ triangle with legs of length $a$, the hypotenuse $= a\sqrt{2}$, by the Pythagorean theorem: $a^2 + a^2 = 2a^2 = (a\sqrt{2})^2$. Key trig values: $\sin(45^\circ) = \cos(45^\circ) = 1/\sqrt{2} = \sqrt{2}/2 \approx 0.707$.
Example
A square with diagonal $10$ has side $= 10/\sqrt{2} = 5\sqrt{2} \approx 7.07$. The diagonal of any square creates two $45$-$45$-$90$ triangles.
Key Insight
Because the two legs are equal, $\sin(45^\circ) = \cos(45^\circ)$. This is the only acute angle where sine equals cosine, making $45^\circ$ a symmetric, special case on the unit circle.
Definition
The $45$-$45$-$90$ triangle is an isosceles right triangle. In the unit circle, it corresponds to the angle $\pi/4$, with terminal point $(\sqrt{2}/2, \sqrt{2}/2)$. This point lies on the line $y = x$, the axis of symmetry that reflects the unit circle to itself, explaining why $\sin(\pi/4) = \cos(\pi/4)$. The side ratio $1:1:\sqrt{2}$ is the simplest instance of the Pythagorean theorem with equal legs.
Example
The eigenvalues of the reflection matrix across $y = x$ are $+1$ and $-1$, with eigenvectors along $(1,1)$ (the $45^\circ$ direction) and $(1,-1)$ (the $135^\circ$ direction). The $45$-$45$-$90$ triangle's geometry is encoded in this reflection symmetry.
Key Insight
In the Eisenstein-Jacobi lattice, the $45$-$45$-$90$ triangle's ratio $\sqrt{2}$ determines the lattice constant. In crystallography, the face-centered cubic structure is based on this ratio, connecting the special triangle to materials science.