Calculate the interior angles of a triangle

Even if angular functions and the consideration of a triangle are compulsory in school mathematics heard, it's not that easy to guess how to calculate the interior angles of a triangle can. However, if you know that in a triangle the sum of all angle is always 180 degrees and knows the right formulas, can solve them correctly and then with the calculator can calculate, it is no longer as difficult as initially feared.

What you need to know about triangles

• The sum of all interior angles of a triangle is always 180 degrees.
• Name the points of the triangle A, B, and C counterclockwise.
• Side a is opposite point A, b opposite B and c opposite C.
• The angles at points A, B, and C are called α, β, γ (alpha, beta, gamma).


Sine law in a non-right-angled triangle - the formula explained using the example

Even in a non-right-angled triangle one can use the trigonometric functions sin and cos ...

This is how you calculate the interior angles of a triangle

1. It is best to first calculate the largest angle that is always opposite the longest side. In this example it should be side a.
2. Since at the beginning you only know the side lengths of the triangle (which you e.g. can measure with a simple ruler), you need the cosine theorem, on one side of which two Sides and the included angle stand, on the other side the side opposite the angle, z. B. a² = b² + c² - 2bc * cos α.
3. Solve this equation for the angle: cos α = (b² + c² - a²) / 2bc. This results in α = arccos ((b² + c² - a²) / 2bc). If you have the Dimensions of a, b, and c, you can enter them all into the calculator.
4. You can of course calculate the other angles in the same way. But since you now know an angle, you can use the 2nd Calculating interior angles is also easier, namely with the help of the sine law. This means that the ratio of the length of one side to the sine of the opposite angle is always the same: i.e. a / sin α = b / sin β = c / sin γ.
5. so you can calculate β: a / sin α = b / sin β. This results in sin β = sin α * b / a, i.e. β = arc sin (α * b / a). Enter everything into the calculator again.
6. You now know two angles; you can easily calculate the last one knowing that all of them three angles together must be 180 degrees by dividing the first two angles from 180 degrees pull off.

While it might seem a bit complicated at first, once you get it, it's actually relatively easy to calculate the interior angles of a triangle.