Triangle Angle Calculator

What this calculator does

So, you have some information about a triangle and you want to fill in the rest, above all its angles. This tool solves a triangle from what you know. Give it three sides, or two angles, or two sides and an angle, and it works out the missing angles, and where relevant the missing sides too.

A dropdown sets which combination you are starting from, and the input boxes change to suit.

How to use it

1. Choose your known combination: three sides, two sides and one angle, or two angles.
2. Enter the values into the boxes that appear.
3. Press Calculate.

For three sides, the tool first checks that the lengths can actually form a triangle before solving.

The rule underneath: angles sum to 180

One fact sits behind everything this tool does: the three interior angles of any triangle always add up to 180 degrees. That single constraint is what makes a triangle solvable. Once you pin down enough of the sides and angles, the rest are forced, because they have to fit both the shape of the triangle and that unbending total of 180. Every mode here is really a way of using this rule together with the right side-and-angle relationship.

From three sides: the law of cosines

When you know all three sides, the tool finds each angle with the law of cosines. This law connects the three side lengths to any one of the angles, so by applying it three times, once per corner, it recovers all three angles from the sides alone. Before it does, it checks the triangle inequality: each side must be shorter than the other two added together, or the lengths simply cannot close into a triangle. If they can, you get all three angles; and reassuringly, they will come out adding to 180.

From two angles: subtraction

This is the simplest case, and it leans directly on the angle-sum rule. If you already know two of the angles, the third is whatever is left after taking them away from 180. The tool checks that your two angles do not already reach or exceed 180, since that would leave no room for a third, then reports the missing angle. No side lengths are needed, because angles alone determine the remaining angle.

From two sides and an angle

The richest mode takes two sides and one angle and reconstructs the whole triangle. Here the tool combines two relationships. It uses the law of sines, which ties each side to the sine of the angle opposite it, to find a second angle, and the law of cosines to pin down the third side. With those in hand, the angle-sum rule delivers the final angle. This mode does more work, but the idea is the same: known pieces plus the standard triangle laws force out the unknown ones.

One thing to know: when you give two sides and an angle that is not between them, there can be two different triangles that fit. This tool reports one of them, so if your problem might have a second, larger-angle answer, it is worth checking for it.

A worked example

Take a triangle with sides 3, 4, and 5, entered in three-sides mode. The lengths pass the triangle inequality, so the tool applies the law of cosines to each corner. It finds angles of 90, about 53.13, and about 36.87 degrees, which add to 180. The 90 tells you this is a right triangle, which fits, since 3, 4, and 5 famously satisfy the Pythagorean relationship.

Questions people ask

What can this tool find?

The angles of a triangle, and the missing sides where the mode calls for it, starting from three sides, two angles, or two sides and an angle.

How does it find angles from three sides?

With the law of cosines, applied once at each corner, after checking that the three lengths can form a triangle.

Why does two angles give the third instantly?

Because the three angles always total 180 degrees, so the third is 180 minus the other two.

When will it refuse to solve?

If three sides break the triangle inequality, or if two given angles already reach 180 and leave no room for a third.

How does this relate to the other triangle tools?

The law of sines calculator focuses on the side-and-angle proportion alone, and the Pythagorean theorem calculator handles the special case of right triangles.

References

On solving triangles. The angles of a triangle sum to 180 degrees, and the law of cosines and law of sines relate its sides to its angles, together determining the unknown parts.

1. Eric W. Weisstein, "Law of Cosines," from MathWorld, a Wolfram resource, on finding an angle from three sides.
2. Eric W. Weisstein, "Law of Sines," from MathWorld, a Wolfram resource, on the proportion between sides and the sines of their opposite angles.