Free Triangle Side Angle Calculator

The Triangle Side Angle Calculator is a free online triangle solver that enables you to determine missing side lengths and angles in any triangle. Whether you need to find a missing side, compute unknown angles, or verify a right triangle, this tool serves as both a triangle side calculator and a triangle angle calculator. It leverages core trigonometric principles such as the Law of Cosines, Law of Sines, and angle sum properties to deliver accurate results.

Key Mathematical Principles

The calculator relies on several fundamental relationships:

• Law of Cosines – Relates the lengths of sides to the cosine of an angle. For a triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$, it is expressed as:

$$c^{2} = a^{2} + b^{2} - 2ab\cos(C)$$

This formula is essential for solving an SSS (side‑side‑side) triangle to find any angle, or an SAS (side‑angle‑side) case to find the third side.

• Law of Sines – States that the ratio of a side’s length to the sine of its opposite angle is constant:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

where $R$ is the circumradius. This is particularly useful when you know one side and its opposite angle along with another side or angle (SSA or AAS configurations).

• Angle Sum Property – The three interior angles always add up to $180^\circ$:

$$A + B + C = 180^\circ$$

• Pythagorean Theorem – A special case of the Law of Cosines that applies when one angle equals $90^\circ$. For a right triangle with legs $a$, $b$ and hypotenuse $c$:

$$a^{2} + b^{2} = c^{2}$$

• Trigonometric Ratios – In right triangles, $\sin \theta = \text{opposite/hypotenuse}$, $\cos \theta = \text{adjacent/hypotenuse}$, and $\tan \theta = \text{opposite/adjacent}$. These allow you to calculate unknown sides or angles given one side and one acute angle.

Supported Input Scenarios

The solver is designed to handle the most common triangle configurations:

1. Three sides (SSS) – The calculator determines all three angles automatically.
2. Two sides and an angle – Depending on whether the angle is included between the sides (SAS) or not (SSA), the missing side and the remaining angles are computed.
3. Two angles (with or without a side) – The third angle follows from the angle sum property, and the Law of Sines provides the remaining side lengths.

When you select the appropriate data‑type option and enter your known values, the tool applies the correct formula or combination of formulas to give you a complete solution.

Worked Example

Suppose you know one angle $\alpha = 30^\circ$ and two sides: $a = 4$ (opposite $\alpha$) and $b = 6$. To solve the triangle:

1. Choose the “Angle and two sides” input mode.
2. Enter $\alpha = 30^\circ$, $a = 4$, and $b = 6$.
3. The calculator first uses the Law of Sines:

$$\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \quad\Rightarrow\quad \frac{4}{\sin 30^\circ} = \frac{6}{\sin \beta}$$

Because $\sin 30^\circ = 0.5$, we have $\frac{4}{0.5} = 8$, so $\sin \beta = \frac{6}{8} = 0.75$. Hence $\beta = \arcsin(0.75) \approx 48.59^\circ$.

4. Using the angle sum property, $\gamma = 180^\circ - 30^\circ - 48.59^\circ = 101.4^\circ$.
5. The missing side $c$ is then obtained from the Law of Sines:

$$c = \frac{a}{\sin \alpha} \cdot \sin \gamma = 8 \cdot \sin 101.4^\circ \approx 8 \cdot 0.9805 = 7.842$$

The output displays side $c = 7.842$, angle $\beta = 48.59^\circ$, and angle $\gamma = 101.4^\circ$.

This example demonstrates how the triangle side angle calculator efficiently processes mixed known data and returns a complete set of results.