Law of Cosines Calculator (Non-Right Triangles)

The Law of Cosines is used to solve any general triangle (non-right triangle) when you know either all three sides (**SSS**) or two sides and the angle between them (**SAS**). Select your known case below.

The Three Forms of the Law of Cosines

The Law of Cosines is a powerful tool because it works for any triangle. It relates the three sides of a triangle to the cosine of one of its angles.

Formulas to Find a Missing Side (SAS Case)

If you know two sides and the included angle (SAS), use one of these formulas to find the third side:

Formulas to Find a Missing Angle (SSS Case)

If you know all three sides (SSS), you can rearrange the primary formula to find any angle:

Law of Cosines vs. Law of Sines

The Law of Cosines is typically the starting point for solving a triangle in the **SAS** and **SSS** cases. Once you have used the Law of Cosines to find one missing value, it is often easier and less ambiguous to use the **Law of Sines** to find the remaining angles.

The Law of Sines (used in ASA, AAS, and SSA cases) is:

The Law of Cosines avoids the "ambiguous case" (SSA), making it the safer choice when applicable.

Frequently Asked Questions (FAQ)

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a^2 = b^2 + c^2 - 2bc \cos(\alpha)

b^2 = a^2 + c^2 - 2ac \cos(\beta)

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

\alpha = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right)

\beta = \cos^{-1} \left( \frac{a^2 + c^2 - b^2}{2ac} \right)

$a^2 = b^2 + c^2 - 2bc \cos(\alpha)$ $b^2 = a^2 + c^2 - 2ac \cos(\beta)$ $c^2 = a^2 + b^2 - 2ab \cos(\gamma)$

$\alpha = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right)$ $\beta = \cos^{-1} \left( \frac{a^2 + c^2 - b^2}{2ac} \right)$ $\gamma = \cos^{-1} \left( \frac{a^2 + b^2 - c^2}{2ab} \right)$

$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$