Right Triangle Calculator

In a right triangle, label the sides relative to the acute angle $\theta$ you care about: the hypotenuse is always opposite the right angle, the opposite side faces $\theta$, and the adjacent side touches it.

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

That is SOH-CAH-TOA. Two more facts complete the toolkit:

$a^2 + b^2 = c^2 \qquad \text{and} \qquad A + B = 90°$

for legs $a, b$, hypotenuse $c$ and acute angles $A, B$.

These ratios only apply to a triangle that actually contains a $90°$ angle. For a non-right triangle you need the Law of Sines or the Law of Cosines instead — SOH-CAH-TOA will give a wrong answer there, not an error message.

You need two pieces of information besides the right angle, and at least one of them must be a side. Match your case:

Case 1: two sides known

1. Find the third side with $a^2 + b^2 = c^2$.
2. Find one acute angle with an inverse ratio, e.g. $A = \arctan\!\left(\frac{a}{b}\right)$ or $A = \arcsin\!\left(\frac{a}{c}\right)$.
3. Get the other angle from $B = 90° - A$.

Case 2: one side and one acute angle known

1. The other acute angle is $90° - A$ immediately.
2. Pick the ratio that connects the side you have to the side you want — that is what "which trig function do I use?" really means. Known hypotenuse and want the opposite side: $a = c\sin A$. Known opposite and want the hypotenuse: $c = a / \sin A$.

Choosing the function

Write down which two sides the problem involves. Opposite and hypotenuse $\Rightarrow$ sine. Adjacent and hypotenuse $\Rightarrow$ cosine. Opposite and adjacent $\Rightarrow$ tangent. Use $\arcsin$, $\arccos$, $\arctan$ when the unknown is the angle.

• Calculator in the wrong mode: degree answers from radian mode are the single most common source of wrong results. Check that $\sin 30° = 0.5$ before you start.
• Fixed labelling: "opposite" and "adjacent" swap when you switch which acute angle you are working from. Only the hypotenuse never moves.
• Using $\sin$ where $\arcsin$ is needed: $\sin$ turns an angle into a ratio; $\arcsin$ turns a ratio back into an angle.
• Applying $a^2 + b^2 = c^2$ with $c$ as a leg: $c$ must be the hypotenuse, the longest side.
• Rounding too early: carry full precision through intermediate steps and round only the final answer, or angles drift by a tenth of a degree.