Free Triangle Degree Calculator

When you need to determine the interior angles of a triangle, the Triangle Angle Calculator (also referred to as a free Geometry Angle Calculator) provides an instant solution. This tool covers all typical data scenarios: when all three side lengths are known, when two interior angles are given, and when two sides together with one angle are available. The underlying calculations rely on the Law of Cosines Calculator algorithm, the Triangle Angle Sum property, and the law of sines.

What Is a Triangle?

A triangle is a three‑sided polygon. Its three interior angles, denoted $\alpha$, $\beta$, and $\gamma$ (for the vertices opposite sides $a$, $b$, and $c$ respectively), always satisfy the angle sum property: $\alpha + \beta + \gamma = 180^\circ$. The triangle can be acute, right, or obtuse depending on its angles, and the calculator automatically classifies the result.

Calculation Methods

The Triangle Degree Calculator uses different mathematical approaches depending on what information you have.

1. Three Sides Are Known (Law of Cosines)

If the lengths of all three sides are provided, the law of cosines is used to find each angle. For side $a$ opposite angle $\alpha$, the relation is:

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

Solving for $\cos(\alpha)$:

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

Applying the inverse cosine gives $\alpha = \arccos\!\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$. The same process is repeated (by cyclically permuting the side labels) to obtain $\beta$ and $\gamma$.

Example – For a triangle with sides $a = 5\ \text{cm}$, $b = 6\ \text{cm}$, $c = 7\ \text{cm}$, the calculator first computes the cosines of the angles and then the angles themselves:

$$\cos(\alpha) = \frac{6^2 + 7^2 - 5^2}{2 \cdot 6 \cdot 7} = \frac{36 + 49 - 25}{84} = \frac{60}{84} = 0.7143,\quad \alpha \approx 44.4^\circ$$ $$\cos(\beta) = \frac{5^2 + 7^2 - 6^2}{2 \cdot 5 \cdot 7} = \frac{25 + 49 - 36}{70} = \frac{38}{70} = 0.5429,\quad \beta \approx 57.1^\circ$$ $$\gamma = 180^\circ - 44.4^\circ - 57.1^\circ = 78.5^\circ$$

2. Two Angles Are Known (Angle Sum)

Thanks to the Triangle Angle Sum theorem, if two angles are known, the third is simply $180^\circ$ minus the sum of the two. For instance, if $\alpha = 80^\circ$ and $\beta = 45^\circ$, then $\gamma = 180^\circ - 80^\circ - 45^\circ = 55^\circ$.

In a right triangle, because one angle is $90^\circ$, the other two acute angles are complementary – their sum equals $90^\circ$. So if one acute angle is known, subtract it from $90^\circ$ to obtain the other acute angle.

3. Two Sides and One Angle Are Known

When you have two side lengths and one angle, the approach depends on the relative position of the known angle.

• SAS (Side‑Angle‑Side): The known angle lies between the two given sides. The law of cosines first yields the missing side, and then the law of cosines (or law of sines) gives the remaining angles.
• SSA (Side‑Side‑Angle): The known angle is opposite one of the given sides. The law of sines is used:

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

From the known side‑angle pair, one can solve for another angle. Note that this case may lead to the ambiguous situation (0, 1, or 2 possible triangles), and the calculator will indicate all valid results.

How to Use the Calculator

1. From the drop‑down menu, select the type of known information: “Three sides”, “Two angles”, or “Two sides & one angle”.
2. Enter the values in the appropriate fields (all sides or angles must be in consistent units).
3. Click Calculate – the tool displays all three interior angles instantly.

Illustration – For a triangle with all sides equal to $4\ \text{cm}$, choose “Three sides”, input $a = b = c = 4$, and the result shows each angle as $60^\circ$. This demonstrates the equilateral property.

The same free Triangle Angle Calculator online works for metric or imperial units for sides and degrees for angles, making it a truly versatile Geometry Angle Calculator.

Conclusion

With its built‑in Law of Cosines Calculator engine and support for the angle sum rule and law of sines, this tool eliminates manual computation errors. Whether you are a student studying geometry, a hobbyist, or a professional, the Triangle Degree Calculator helps you calculate triangle angles quickly and accurately.