Angle Calculator (Right Triangle Solver)
This calculator solves for the missing angles and sides of any right-angled triangle. Enter the minimum two known values (two sides, or one side and one acute angle) in the corresponding fields below.
Enter Known Values (Minimum 2)
Angle $\gamma = 90^\circ$
Solved Triangle
Side Results
Angle Results
Step-by-Step Solution
The Core Formulas of Right Triangle Trigonometry
A right triangle is fully defined by three key relationships:
1. Pythagorean Theorem (Sides)
Used to find the length of any side when the other two sides are known:
Where $c$ is the hypotenuse (the side opposite the $90^\circ$ angle), and $a$ and $b$ are the legs.
2. Angle Sum Rule
The sum of the internal angles of any triangle is $180^\circ$ (or $\pi$ radians). Since the right angle $\gamma$ is $90^\circ$, the two acute angles ($\alpha$ and $\beta$) must sum to $90^\circ$:
3. SOHCAHTOA (Sides and Angles)
The trigonometric ratios relate the angles to the ratio of side lengths:
$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$
To find a missing angle ($\theta$) when all sides are known, we use the inverse functions (arcsin, arccos, arctan): $\theta = \sin^{-1}(\frac{O}{H})$.