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.

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:

To find a missing angle ($\theta$) when all sides are known, we use the inverse functions (arcsin, arccos, arctan): $\theta = \sin^{-1}(\frac{O}{H})$.

Frequently Asked Questions (FAQ)

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a^2 + b^2 = c^2

\alpha + \beta + 90^\circ = 180^\circ \quad \text{or} \quad \alpha + \beta = 90^\circ

\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

c^2 = a^2 + b^2

\tan(\alpha) = \frac{a}{b}

$\alpha + \beta + 90^\circ = 180^\circ \quad \text{or} \quad \alpha + \beta = 90^\circ$

$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$