How To Figure Out The Third Side Of A Triangle

You've got two sides and the angle between them. Day to day, or maybe two angles and a side. Or — and this is the one that trips people up — you only know two sides and the angle isn't the one you need. How do you find the missing side of a triangle?

Turns out, it depends entirely on what you already have. There's no single "triangle side formula" that works for every case, and that's where most guides let you down. They hand you the Law of Cosines and call it a day. But the real question is: which tool do you reach for, and when?

What "Finding the Third Side" Actually Means

A triangle has three sides and three interior angles that always add up to 180 degrees. If you know any three of these six values (as long as one of them is a side), you can figure out the rest. This is called solving the triangle.

The tricky part is that "knowing three things" can show up in a few different ways, and each one has its own method:

• SAS — two sides and the included angle (the angle between those two sides)
• SSS — all three sides (but then you're not looking for a side, you're verifying the triangle exists)
• SSA — two sides and a non-included angle (the ambiguous case, more on this below)
• AAS or ASA — two angles and a side, which lets you find the third angle first

Knowing which case you're in is half the battle. The other half is picking the right formula.

Why People Get Stuck on This

The confusion usually comes from one of two places. Either someone hands you a problem without telling you which case you're in, or you vaguely remember the Law of Cosines and try to force it onto a problem that actually calls for something simpler.

Another common trap: the Law of Sines and the Law of Cosines look similar on paper, and a lot of students mix them up or use them in the wrong situation. The Law of Sines is a ratio. The Law of Cosines looks like a quadratic. They behave differently and they answer different questions.

And then there's the ambiguous case — SSA — which can produce zero, one, or two valid triangles. Still, most beginners don't even know that part exists, so when they "get the right answer" using a formula and it turns out to be wrong, they assume they're bad at math. This leads to they're not. They just got a triangle that lies.

How to Find the Third Side: The Three Main Methods

When You Have Two Sides and the Included Angle (SAS)

This is the Law of Cosines situation. The formula looks like this:

c² = a² + b² − 2ab·cos(C)

Where C is the angle between sides a and b, and c is the side opposite that angle — which is the side you're trying to find.

Quick example. Say a = 7, b = 10, and the angle between them is 60 degrees. Then:

c² = 49 + 100 − 2(7)(10)·cos(60°) c² = 149 − 140·(0.5) c² = 149 − 70 c² = 79 c ≈ 8.89

That's it. When the angle is 90 degrees, the cos(90°) term drops to zero, and you're left with the Pythagorean theorem. Here's the thing — you take the square root at the end. This formula works for any SAS triangle, no matter what the angle is, including obtuse ones. Which makes sense — the Law of Cosines is basically the Pythagorean theorem with a correction term.

When You Have Two Angles and a Side (AAS or ASA)

Here, the Law of Sines is your friend. The formula is:

a / sin(A) = b / sin(B) = c / sin(C)

You already know one side and its opposite angle. From there, you can find the third angle by subtracting the two known angles from 180. Then you set up a ratio.

Say you know angle A = 40°, angle B = 75°, and side a = 12 (side a is opposite angle A). First, angle C = 180° − 40° − 75° = 65°.

Now set up the ratio:

12 / sin(40°) = c / sin(65°)

c = 12 · sin(65°) / sin(40°) c ≈ 12 · 0.9063 / 0.6428 c ≈ 16.

Same idea for finding side b. This method feels cleaner because you're not dealing with squares and square roots, just ratios.

When You Have Two Sides and a Non-Included Angle (SSA)

This is the one that makes people want to throw their textbook out the window. It's called the ambiguous case because depending on the values, you can get:

• No triangle (the given side is too short to reach the base)
• One triangle (right triangle, or the given side is just long enough)
• Two triangles (the given side can swing and hit the base in two places)

You start with the Law of Sines, but then you have to check whether a valid triangle even exists.

Here's the quick check. If you know sides a and b and angle A (where A is opposite side a), compute:

sin(B) = b · sin(A) / a

Then look at the value:

• If sin(B) > 1, no triangle exists
• If sin(B) = 1, exactly one triangle (a right triangle)
• If sin(B) < 1, you get two possible values for angle B (B and 180° − B), but you have to check whether both actually work. If A + B ≥ 180°, one of them gets thrown out.

Honestly, this is the case that deserves the most caution. Even experienced students get caught by it.

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Common Mistakes People Make

Forcing the wrong formula. If you know two sides and the angle between them, don't reach for the Law of Sines. It won't work because you don't have a side-angle pair. The Law of Cosines needs the included angle specifically.

Forgetting the square root. With the Law of Cosines, you square the unknown side, so the last step is always taking a square root. It's easy to leave the answer squared, especially under time pressure.

Mixing up which angle goes with which side. In the Law of Sines, the side and angle have to be opposite each other. If they're not, the ratio breaks.

Ignoring the ambiguous case. SSA problems can secretly have two answers, or no answer at all. Always check.

Rounding too early. If you're working through multiple steps, keep at least four or five decimal places until the very end. Rounding at each step compounds the error, and your final answer can drift by a noticeable amount.

Assuming the answer should be "nice". Triangle problems in real life (engineering, surveying, navigation) don't give you clean numbers. If your answer is something like 11.347, that's normal.

Practical Tips That Actually Help

Draw the triangle first. Even a rough sketch. Label what you know. This single habit prevents most of the formula-mixup errors people make. You can literally see which angle is between which sides.

Identify your case before you compute anything. SAS? SSS? ASA? AAS? SSA? Spend five seconds naming it. Then pick the right tool.

Keep your calculator in the right mode. If your calculator is in radians but your angle is in degrees, your answer will be wildly off. This is a classic and embarrassing mistake — almost everyone has done it at least once.

Sanity-check the answer. The third side of a triangle should be smaller than the sum of the other two sides and larger than the absolute difference. If your answer violates that, something went wrong.

For the Pythagorean theorem, don't reach for it unless the angle is 90 degrees. It's tempting because the formula is so familiar, but a triangle with a 60-degree angle isn't right-angled, and the Pythagorean theorem will give you the wrong answer.

Use the Law of Cosines when the angle is the unknown and the Law of Sines when a side is the unknown. That's a decent rule of thumb. The Law of Cosines is also the only one of the two that can give you all three angles when you know all three sides.

FAQ

What if I know all three sides but no angles?

You don't need to "find" a side,

you need to find angles. Then use the inverse cosine function on your calculator. The Law of Cosines is your tool here. Pick any one angle, plug the three known sides into the Law of Cosines, and solve for its cosine. Repeat for the other two angles, or just use the Law of Sines once you have one angle.

Can I use both laws in the same problem?

Absolutely, and often you should. A common pattern: use the Law of Cosines to find one unknown side when you have SAS or SSS, then switch to the Law of Sines to find one of the remaining angles more quickly (it's usually less algebra than running Law of Cosines again).

What about the ambiguous case in more detail?

When you have SSA, you compute the height of the possible triangle using h = b·sin(A). Worth adding: if a (the side opposite A) is less than h, no triangle exists. If a equals h, exactly one right triangle exists. That said, if a is greater than h but less than b, two valid triangles exist. If a is greater than or equal to b, one triangle exists.

Do these laws work in three dimensions?

Yes. In practice, if you have a triangle drawn in 3D space, the law of sines and law of cosines still hold for that triangle. Which means the "angle between two sides" still means the interior angle of the triangle, not the angle between the lines extending in 3D space. The only thing 3D adds is figuring out the side lengths and interior angles in the first place, which usually requires vector projections.

What if my problem has no triangle at all, but a shape made of triangles?

Break it up. Find the triangles inside the figure, solve each one separately, and combine the results. Many quadrilateral problems, for instance, are really just two triangle problems stuck together by a diagonal.

Is there a formula for the area?

Two of them, actually. The SAS area formula is (1/2)·a·b·sin(C), which uses two sides and the included angle. Heron's formula gives you the area from the three side lengths: area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (a+b+c)/2. Both are worth knowing.

Why does the Law of Cosines have a minus sign while the Pythagorean theorem doesn't?

About the Py —thagorean theorem is a special case of the Law of Cosines where the angle C is 90 degrees. Day to day, since cos(90°) = 0, the "-2ab·cos(C)" term drops out, and you get a² + b² = c². The minus sign is there so the formula works for all angles, not just right ones.

Wrapping Up

The Law of Sines and the Law of Cosines are the workhorses of triangle trigonometry. Now, when you've got an angle sandwiched between two known sides, or all three sides and no angles, reach for the Law of Cosines. And knowing which one to use for a given problem is half the battle. When you've got a side-angle pair and need to find something across the triangle, reach for the Law of Sines.

Most errors come from rushing: grabbing the wrong formula, rounding too early, mixing up which angle goes where. Slow down for ten seconds, draw the triangle, identify the case, and pick the right tool. Your accuracy will jump.

The rest is just careful arithmetic and a well-set calculator.