Find The Third Side Of A Triangle

How Do You Find the Third Side of a Triangle?

You're staring at a triangle on your math worksheet — or maybe it's a real-world problem, like figuring out how long a fence needs to be, or how far apart two points are on a map. The third one is missing. Two sides are given. So what now?

Here's the good news: there's almost always a way to find it, and you probably already know more than you think. So the trick is figuring out which* method matches the information you've got. Let me walk you through the real, practical ways to do it — no robotic steps, no "first we must understand" filler. Just the math, the shortcuts, and the stuff that actually comes up.

What You're Really Working With

A triangle has three sides, three angles, and a handful of useful relationships baked in by geometry. The "third side" problem usually comes in a few flavors:

• You've got two sides and the angle between them.
• You've got two sides and an angle that isn't between them.
• You've got one side and two angles.
• You've got all three sides (and you're trying to check* if the triangle even exists).

Each flavor has its own tool. And the tool you pick is everything.

The Law of Cosines — Your Most Reliable Friend

If you've got two sides and the included angle (the angle between* them), the Law of Cosines is almost certainly what you want. It works for any triangle, not just right ones, which is why it's so handy.

The formula looks like this:

c² = a² + b² − 2ab·cos(C)

In plain English: square the two known sides, add them, then subtract twice their product multiplied by the cosine of the angle between them. Take the square root of whatever you get, and there's your third side.

Let's say you've got sides of 7 and 10, with an angle of 60° between them. You'd plug in:

c² = 49 + 100 − 2(7)(10)(0.5) c² = 149 − 70 c² = 79 c ≈ 8.89

That's it. The third side is about 8.89.

Why this works when other things don't

The Pythagorean theorem is basically the Law of Cosines in disguise — it just assumes the angle is 90°, which makes the cosine term drop out. So when people hit a non-right triangle and Pythagoras "doesn't work," it's not that the math is broken. They just need the more general version.

Honestly, if you remember one formula for this whole topic, make it this one. It handles the widest range of problems.

The Law of Sines — When You've Got an Angle and a Side Across From It

Different situation: you know one side, the angle right across from it, and one other* angle. Now you can find the side across from that other angle using the Law of Sines:

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

You rearrange it to solve for the missing side. This is especially useful in surveying, navigation, and physics problems where you can measure an angle but not a length.

One thing to watch out for: this method can give you two possible answers (the famous "ambiguous case"). If that happens, the extra information — like which side is longer — usually tells you which answer is real.

Right Triangles Get a Shortcut

If you're lucky enough to have a right triangle (one 90° angle), things simplify fast. Two famous rules come into play:

• Pythagorean theorem for the sides themselves: a² + b² = c², where c is the hypotenuse.
• SOH-CAH-TOA if you're hunting for a side based on an angle.

If the right angle is between the two known sides, you're looking at the hypotenuse. Day to day, just square both, add, take the square root. If the right angle is somewhere else, you might be looking at a leg — and the hypotenuse becomes one of your known values.

For a quick example: legs of 5 and 12. The hypotenuse is √(25 + 144) = √169 = 13. Which means classic 5-12-13 triangle. Done.

What Most People Get Wrong

Mixing up which angle goes where

This is the big one. So naturally, with the Law of Cosines, the angle you use must* be the one between the two known sides. If you grab a random angle, the answer will be wrong and you won't immediately know why.

A good habit: draw the triangle, label everything, and actually look at the angle. Consider this: no? Think about it: use it. Consider this: is it sitting between the two sides you know? And yes? You probably need a different formula. That's the whole idea.

Forgetting the square root

Let's talk about the Law of Cosines gives you c², not c. I've seen students carefully compute everything correctly, write down the squared value, and then stop. The answer is one more step away. Take the square root.

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Ignoring whether the triangle is even possible

The Triangle Inequality Theorem says the sum of any two sides must be greater than the third. On the flip side, if your known sides are 2 and 3, the third side can never be 6 — the triangle wouldn't close. Worth a quick check before you sink time into a calculation that has no real answer.

Rounding too early

If you're working through a multi-step problem, don't round your intermediate values. Carry extra digits and only round at the end. Plus, rounding halfway through is how answers drift from 8. 88 to 8.94, and in some classes or professions, that error actually matters.

Practical Tips That Actually Help

Sketch the triangle, even if it's rough

You don't need graph paper. A quick freehand sketch with the sides and angles labeled is often enough to spot which formula fits. It also helps you catch obvious mistakes — like accidentally writing down a 40° angle where the picture clearly shows 140°.

Decide on the formula before* you plug anything in

Look at what you're given. Two sides and an included angle? And law of Cosines. In practice, two angles and a side? That said, law of Sines. Day to day, a right angle anywhere? That said, pythagoras or trig ratios. Picking the right formula first saves you from going down the wrong path entirely.

Keep your calculator in the right mode

If your calculator is in radians when you need degrees, your cosine values will be way off and your "answer" won't make any sense. This is a small thing that ruins a lot of homework problems. Double-check it.

For real-world problems, watch your units

If a side is in meters and the angle is in degrees, your answer will be in meters. But mixing units — kilometers and meters in the same calculation — is a classic source of answers that are off by a factor of 1,000. Fine. Convert everything to one unit before you start. Surprisingly effective.

When in doubt, work backward

Plug your answer back into the original equation. Does it hold? Does it match the picture? Quick sanity checks catch most errors before anyone else does.

FAQ

What if I know all three sides but not any angles?

That's actually a different problem — you're computing an angle, not a side. The Law of Cosines handles this too: solve for cos(C), then take the inverse cosine to get the angle itself.

Can I find the third side with no angles at all?

No. If you only know two sides, there are infinitely many triangles that could fit them (imagine hinging a triangle at one vertex and swinging the third side around). You need something* — usually an angle, sometimes a relationship between sides — to lock it down.

What if my angle is opposite the missing side?

That's a slightly trickier case. The Law of Cosines can still handle it if you know the other two angles, but you'll need to use the Law of Sines first to find an additional side, then come back to Cosines. Two-step, but doable.

Is there a formula for an equilateral triangle?

If all sides are equal, the "third side" is just the same as the other two. For an equilateral triangle with side s, the height is (s·√3)/2 if you ever need it, but the sides themselves are trivial.

What tools can check my work?

Any scientific calculator with cosine and inverse cosine functions will do the job. Online triangle solvers are also handy for verifying your answer, but they're no substitute for knowing the method — because in a test or on the job, you won't have one.

Wrapping Up

Wrapping Up

Mastering the art of finding a third side in a triangle comes down to understanding your tools and knowing when to use each one. Whether you're applying the Law of Cosines, reaching for the Law of Sines, or simply using basic trigonometric ratios, the process becomes intuitive once you recognize the patterns that different problem types create.

The key takeaways are straightforward: always assess what information you have before choosing a method, keep your calculator settings and units consistent, and verify your work whenever possible. These habits will serve you well beyond triangle problems—in fact, they'll improve your mathematical reasoning overall.

Practice is what transforms these steps from memorized procedures into genuine understanding. Day to day, each triangle you solve reinforces the relationships between sides and angles, building intuition that makes harder problems feel manageable. Start with the straightforward cases, then gradually challenge yourself with ambiguous cases (SSA) and real-world applications.

Remember, every triangle problem has a solution. Practically speaking, the methods exist, the formulas work, and with careful attention to detail, you can find that third side every time. Now you have the tools and the knowledge—all that's left is to put them to use.