Triangle angle calculator

What are the angles of a triangle?

Triangle angles are the angles formed by the two sides of a triangle. Every triangle has three angles, and the sum of these angles is always 180 degrees. The angles can be denoted as α (alpha), β (beta), and γ (gamma).

The triangle angle calculator is an online tool that allows you to calculate the angles of a triangle based on known information about other angles and sides. Triangles are a fundamental geometric shape, and understanding their angles and sides is important in both theoretical mathematics and practical applications like architecture and engineering design.

Properties of triangle angles

1. Sum of angles: As mentioned earlier, the sum of all three angles of any triangle is always 180 degrees.
2. Depending on the angles, a triangle may be:
   • Acute-angled, if all angles are less than 90 degrees.
   • Right-angled, if one of the angles is 90 degrees.
   • Obtuse-angled, if one of the angles is greater than 90 degrees.

Formulas

The calculation of triangle angles depends on the known data. If two angles are known, the general rule of the sum of all triangles is used; when the lengths of all sides are known, the cosine theorem should be used, and if two sides and the angle between them are known - the sine theorem. Let’s break down each of the calculation options:

Sum of all angles

A triangle has an important property: the sum of its interior angles is always 180 degrees. This fundamental property follows from Euclidean geometry and is the basis for many other geometric calculations.

When two angles are initially known, the third angle can always be calculated from the equation:

$$\gamma = 180^\circ - \alpha - \beta$$

This rule simplifies solving many tasks related to triangles and represents a basic property that can be used for quickly finding unknown angles.

Cosine theorem

The cosine theorem allows you to calculate angles if the lengths of all three sides of a triangle are known. It states that the square of the length of any side of a triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of these sides multiplied by the cosine of the angle between them. Formulas for calculating angles using the cosine theorem:

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

After finding the cosine of an angle, you can use the arccos function to find the angle itself.

Sine theorem

To calculate angles with two known sides and the angle between them, you can use the sine theorem. It states that the ratio of the length of a side to the sine of the opposite angle is the same for all three sides of the triangle:

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

Examples

Example 1: Calculating an angle with two known angles

Suppose we have a triangle where $\alpha = 50^\circ$ and $\beta = 60^\circ$. Then angle $\gamma$:

$$\gamma = 180^\circ - 50^\circ - 60^\circ = 70^\circ$$

Example 2: Calculating an angle with three sides

Consider a triangle with sides $a = 7$, $b = 10$, $c = 5$. Calculate angle α:

$$\cos(\alpha) = \frac{10^2 + 5^2 - 7^2}{2 \cdot 10 \cdot 5} = \frac{100 + 25 - 49}{100} = \frac{76}{100} = 0.76$$

Now find angle α:

$$\alpha = \arccos(0.76) \approx 40.54^\circ$$

Example 3: Calculating angles with two sides and angle between them

Suppose sides $a = 6$, $b = 8$, and the angle between them $\alpha = 45^\circ$ are known. Then to find angle β:

$$\frac{6}{\sin(45^\circ)} = \frac{8}{\sin(\beta)}$$

Solve for $\sin(\beta)$:

$$\sin(\beta) = \frac{8 \cdot \sin(45^\circ)}{6} = \frac{8 \cdot \frac{\sqrt{2}}{2}}{6} = \frac{4\sqrt{2}}{6} = \frac{2\sqrt{2}}{3}$$

Find angle β:

$$\beta = \arcsin\left(\frac{2\sqrt{2}}{3}\right) \approx 73.74^\circ$$

Notes

1. When using arccos and arcsin, ensure the results are within the permissible range of angles (0-180 degrees).
2. In cases where a triangle cannot be formed with the given parameters, results may not match real angle values.
3. Ensure the input data is correct and allowable for triangle construction, as incorrect data will lead to calculation errors.

Frequently asked questions

How to find the third angle of a triangle if two angles are given?

If two angles $\alpha$ and $\beta$ are known, the third angle $\gamma$ can be found by the formula:

$$\gamma = 180^\circ - \alpha - \beta$$

How are angles calculated if three sides of a triangle are known?

To find angles when three sides are known, the cosine theorem is used. Using the formula:

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

and arccos to find angle α.

What to do if angle calculation is impossible?

If calculation is impossible (e.g., the sides violate the triangle inequality), recheck the entered data. It is possible that such parameters cannot form a triangle.

Triangle $abc$, how to find angle $\angle ac$?

If the sides of the triangle are $a, b$, and $c$, to find angle $\angle ac$, apply the following calculations:

Use the cosine theorem to calculate angle $\gamma$:

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

After calculating $\cos(\gamma)$, use arccos to find the angle $\gamma$ itself:

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

Can this calculator be used for right triangles?

Yes, the calculator is also suitable for right triangles. For known hypotenuse and one leg, you can find one of the angles using trigonometric functions.

In a triangle, the angle is 90 degrees, how to find the other angles?

If one angle of a right triangle is 90 degrees, besides this calculator, you can also use a special right triangle angle calculator.