# How do I find the angles of a triangle if I only have the lengths of the sides?

Is it possible to find the angles of a triangle if I only have its sides? If so, how can I achieve this?

Regarding my knowledge of triangles: Whilst I was taught trigonometry a few years ago, I cannot for the life of me remember how to do things like use SOHCAHTOA to figure out the length of a side given an angle and a side. I know it's possible and if that were my problem I would continue searching the internet for a solution, but I gather finding an angle without knowing any of the angles is more difficult.

Any input is appreciated, and I thank you for your time in advance.

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If you want to learn it then you need to study the Law of Cosines. If you need a formula then it is $\cos A={{b^2+c^2-a^2}\over {2ab}}$ – Maesumi Feb 2 '13 at 17:56
is that (2*a)*b or 2*(b*a)? – Pharap Feb 2 '13 at 18:35
Sorry to correct the formula I wrote it is $\cos(A)={{b^2+c^2-a^2}\over {2*b*c}}$. In the denominator either $(2*b)*c$ or $2*(b*c)$ will be same. – Maesumi Feb 2 '13 at 19:36

Use the Cosine Law.

Let $\triangle ABC$ have sides $a$, $b$, and $c$. We are using the usual convention that the length of the side opposite vertex $A$ is called $a$, and so on.

Let $\theta=\angle C$. Then the Cosine Law says that $$c^2=a^2+b^2-2ab\cos \theta.$$ Since we know $a$, $b$, and $c$, we can use the above formula to calculate $\cos\theta$. Then we can use the $\cos^{-1}$ button on the calculator to find $\theta$ to excellent accuracy.

We can use the Cosine Law three times to get the three angles. But we only need to do the calculation for two of the angles: If we have them, the third can be easily found.

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Once you have one angle, it’s a bit quicker to use the Law of Sines for the other angles. By the way, I recommend to high-school students to find the largest angle first, for then the ambiguity in using L of S for angles does not occur. – Lubin May 1 '14 at 17:20
Does this work the same for either obtuse or acute triangle? – Doug Null Sep 26 '15 at 12:15
The Cosine Law will always work, for all the angles, though it would be unreasonable to use it for all three, since once we know two the third is easy. Lubin suggested using the Cosine Law for the first calculation and the Sine Law for the second. This is faster than Cosine Law done twice. The suggestion is to use the Cosine Law to determine the angle opposite a largest side. Then we will know that the other angles are acute, so Sine Law goes smoothly. This works for all triangle types. – André Nicolas Sep 26 '15 at 12:42

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