Special Right Triangles Explorer

This is an isosceles right triangle. The two acute angles are both 45°, so the two legs are equal. The hypotenuse is always the leg times √2.

If a leg has length $x$, then $\text{hyp} = x\sqrt{2}$. Going the other way, a leg is $\dfrac{\text{hyp}}{\sqrt{2}}$, which is the hypotenuse times $\dfrac{\sqrt{2}}{2}$.

This triangle is half of an equilateral triangle. The shortest side sits opposite the 30° angle, the longer leg opposite the 60° angle, and the hypotenuse opposite the right angle.

With short leg $x$, the long leg is $x\sqrt{3}$ and the hypotenuse is $2x$. The hypotenuse is exactly twice the short leg, which is the fastest fact to use.

These two triangles appear so often that their ratios are worth memorizing. Knowing them lets you find every side without a calculator and keep answers in exact radical form.

Exact forms are also what you need for the unit circle. The 45-45-90 triangle gives $\sin 45^\circ = \dfrac{\sqrt{2}}{2}$, and the 30-60-90 triangle gives $\sin 30^\circ = \dfrac{1}{2}$ and $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$.

Choose the triangle type, then choose which single side you know. For a 45-45-90 triangle you can start from a leg or the hypotenuse. For a 30-60-90 triangle you can start from the short leg, long leg, or hypotenuse.

Enter the length and the tool fills in the remaining sides, showing each one as a multiple of the base length and as a decimal approximation. The diagram updates to scale, with the right angle marked and the 45/45 or 30/60 angles labeled. Use the Share Link button to save a configuration.