How to Find All Three Angles When You Know All Three Sides

Three side lengths pin a triangle down completely. There is exactly one shape they can make, so the angles are already decided and only need extracting. The cosine rule is the tool that does it, and the only real decision is the order you work in.

Why the largest angle goes first

It is tempting to find one angle with the cosine rule and then switch to the sine rule, which looks like less arithmetic. That shortcut is where people lose marks, because sine cannot tell an acute angle from an obtuse one. Both 130 degrees and 50 degrees have a sine of about 0.766, and a calculator asked for the inverse sine always hands back the acute one.

Take a triangle with sides of 5, 6 and 10. The angle facing the 10 is genuinely 130.54 degrees, but its sine is 0.7599, and asking a calculator for the inverse sine of that returns 49.46 degrees. That is a real number belonging to nothing in this triangle, and nothing on the page flags it.

Two habits avoid this entirely. Either use the cosine rule for every angle, since it returns obtuse angles correctly as a negative cosine, or find the largest angle first with the cosine rule. Only one angle in a triangle can exceed 90 degrees, and it always faces the longest side, so once it is out of the way everything remaining is acute and safe.

A worked example

Take the 13, 14, 15 triangle. The longest side is 15, so start with the angle facing it. The cosine rule gives 13 squared plus 14 squared minus 15 squared, over 2 times 13 times 14, which is 140 over 364, or 0.3846. The inverse cosine of that is 67.38 degrees.

Now the angle facing 14. That works out as a cosine of 0.5077, giving 59.49 degrees. The last angle needs no formula at all: 180 minus 67.38 minus 59.49 leaves 53.13 degrees.

The three angles sum to exactly 180, which is the check. If yours lands on 179.9 that is rounding, but anything further off means an arithmetic slip worth finding.

Largest angle, longest side

In every triangle the angles come in the same order as the sides they face. The largest angle sits opposite the longest side and the smallest opposite the shortest. In the worked example the sides run 13, 14, 15 and the angles run 53.13, 59.49, 67.38 in exactly that order.

This is worth a glance at the end of any calculation. If your largest angle is not facing your longest side, two values have been paired up wrongly somewhere, and the sum can still come to 180 while the labels are swapped.

When the triangle turns out to be right angled

If one of the cosines comes out as exactly zero, that angle is 90 degrees and you have a right triangle. It falls out of the formula naturally, because a cosine of zero means the numerator was zero, which means the two shorter sides squared added to the longest squared. That is Pythagoras appearing as a special case of the cosine rule.

Mistakes that catch people out

• Using the sine rule for the second angle. It cannot distinguish acute from obtuse, and it returns the acute answer either way.
• Pairing an angle with the wrong side. Each angle faces the side of the same name. Check that the largest ends up opposite the longest.
• Treating a negative cosine as an error. It means the angle is obtuse, which is a normal result the formula handles on its own.
• Rounding partway through. Carry the cosine to at least four places. Rounding it before the inverse loses more than you would expect.
• Leaving the calculator in radians. Test with the inverse cosine of zero, which should return 90 in degree mode.

Practice questions

Common questions

How do I find the angles of a triangle from its three sides?

Use the cosine rule. For the angle facing side a, take b squared plus c squared minus a squared, divide by 2bc, then take the inverse cosine.

Which angle should I calculate first?

The one facing the longest side. It is the largest angle and the only one that could be obtuse, so getting it out of the way first makes the rest safe.

Why not use the sine rule for the second angle?

Because sine cannot distinguish an acute angle from an obtuse one. A calculator asked for an inverse sine always returns the acute answer, which may be the wrong one.

What does a negative cosine mean?

The angle is obtuse, meaning greater than 90 degrees. This is not an error, and the inverse cosine handles it correctly on its own.

Do the angles have to add to 180?

Yes, in flat geometry they always do. Use it as your check. A small gap is rounding, but a large one means a mistake in the working.

What if the three sides cannot form a triangle?

The cosine rule returns a value outside the range of minus one to one and the inverse cosine fails. Test the lengths with the triangle inequality first.

Related lessons and calculators

• How to Tell if Three Lengths Can Actually Form a Triangle worth running before this, since impossible lengths break the formula.
• How to Find the Area of a Triangle From Three Sides for the area from the same three measurements.
• How to Find the Area of a Triangle From Two Sides and the Included Angle for using one of these angles once you have it.
• Triangle Solver which runs the cosine rule for all three angles at once.
• Area of a Triangle: Every Formula and When to Use Each for the overview of every area method in one place.
• National Council of Teachers of Mathematics for curriculum standards and how this topic is sequenced in school.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.