Free Triangle Length Calculator

What This Triangle Solver Does

A triangle length calculator—often called a missing side of triangle calculator—determines the unknown side length when you supply two side lengths and the included angle. This free online triangle solver relies on the Law of Cosines, making it equally effective for right, acute, and obtuse triangles.

Understanding the Law of Cosines

The interior angles of a triangle always sum to $180^{\circ}$, but that property alone does not relate side lengths. The Law of Cosines fills that gap:

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

where $a$ and $b$ are the known sides, $\gamma$ is the angle between them, and $c$ is the opposite (unknown) side. The same formula can be rearranged for any pair of sides and their included angle.

How to Use the Calculator

1. Enter the length of the first side ($a$).
2. Enter the length of the second side ($b$).
3. Specify the included angle ($\gamma$) in degrees or radians (depending on the calculator’s setting).
4. The tool instantly returns the length of the third side ($c$).

If all three sides are already known, the equation can be solved for any angle:

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

Thus, the triangle side length calculator also works as an angle finder when three sides are provided.

Example Calculations

Right triangle
A right triangle has legs of 3 and 4 units and a right angle between them ($\gamma = 90^{\circ}$). Because $\cos(90^{\circ}) = 0$, the Law of Cosines simplifies to the Pythagorean theorem:

$$c = \sqrt{3^{2} + 4^{2}} = \sqrt{9 + 16} = 5$$

The calculator outputs a third side of 5 units.

Non‑right triangle
Consider a triangle with sides $a = 7$ and $b = 9$ and an included angle $\gamma = 120^{\circ}$. The Law of Cosines gives:

$$c^{2} = 7^{2} + 9^{2} - 2 \cdot 7 \cdot 9 \cdot \cos(120^{\circ}) = 49 + 81 - 126 \cdot (-0.5) = 130 + 63 = 193$$

Hence, $c \approx 13.89$ units. This shows the law of cosines calculator handles obtuse angles just as well as acute ones.

Key Benefits of This Approach

Because it uses the Law of Cosines, this missing side of triangle calculator directly handles the SAS (side‑angle‑side) case without needing to first compute an angle via the Law of Sines. The result is fast, reliable, and applicable to a wide range of geometry, physics, and engineering problems.

What It Covers (and What It Does Not)

The tool is optimized for the SAS configuration. If your known values are two sides and a non‑included angle (SSA), the Law of Cosines cannot be applied directly, and a different triangle solver that accounts for the ambiguous case would be needed. For the common SAS scenario, however, this calculator is the simplest way to find triangle side length online.