How To Determine The Third Side Of A Triangle

You know the two sides. You know the angle between them. So what is the third side?

That's the question that trips up more people than it should. Not because the math is hard, but because most explanations jump straight to the Law of Cosines without ever explaining why it works, when to use it, and what to do when the situation you're actually facing doesn't quite match the textbook example.

If you've ever stared at a triangle problem and felt that little flicker of uncertainty — "is this the right formula? That's why did I set it up right? " — this guide is for you.

What "Finding the Third Side" Actually Means

In geometry, a triangle has three sides and three angles, and if you know enough about any three of those six things, you can figure out the rest. Finding the third side usually means you're working in one of these situations:

SAS (Side-Angle-Side): You know two sides and the angle between them.
SSS (Side-Side-Side): You know all three sides, and you're trying to find an angle.
SSA (Side-Side-Angle): You know two sides and an angle not between them — the tricky one.
AAS or ASA: You know angles and a side, and you need another side.

The most common scenario, and the one most people mean by "find the third side," is the SAS case. And the tool for that job is the Law of Cosines.

Why the Pythagorean Theorem Isn't Enough

If you've been in a math class anytime in the last two thousand years, you know the Pythagorean theorem: a² + b² = c². Clean. That said, beautiful. Works perfectly — for right triangles only*.

A right triangle has one 90° angle, and that fixed angle is what makes everything simplify so nicely. The moment your triangle isn't a right triangle, the Pythagorean theorem stops being accurate.

So if someone hands you a triangle with two sides of 7 and 10 and an angle of 60° between them, you cannot just say √(7² + 10²) and call it a day. That answer will be wrong.

This is where the Law of Cosines comes in. Which means it's the Pythagorean theorem's more flexible cousin. It works on every* triangle, not just the right-angled ones.

The Law of Cosines, Explained Like a Person

Here's the formula:

c² = a² + b² − 2ab·cos(C)

Where:

a and b are the two known sides
C is the angle between them (the included angle)
c is the side opposite that angle — the one you want to find

Notice what happens if C = 90°. In practice, that's the Pythagorean theorem. Plus, the cosine of 90° is 0, so the entire "−2ab·cos(C)" term drops out, and the formula collapses to c² = a² + b². Same formula, just a special case.

That's actually one of the nicest things about the Law of Cosines — it doesn't replace what you already know. It contains it.

Walking Through a Real Example

Let's say you have a triangle with sides a = 6 and b = 8, and the angle C between them is 50°.

Step 1: Square the known sides.

a² = 36
b² = 64

Step 2: Calculate the cosine term. 6428

2ab = 2 × 6 × 8 = 96
2ab·cos(C) = 96 × 0.Think about it: - cos(50°) ≈ 0. 6428 ≈ 61.

Step 3: Plug it in.

c² = 36 + 64 − 61.71 ≈ 38.

Step 4: Take the square root.

c ≈ √38.29 ≈ 6.

So the third side is about 6.19 units long. Done.

If the angle had been 90°, the answer would be √(36 + 64) = √100 = 10. That said, makes sense — bigger angle between the sides, longer the opposite side. That's a useful sanity check to build into your head: as the included angle gets bigger, the opposite side gets longer.

When the Angle You Know Isn't Between the Two Sides

This is the case that derails people. SSA — you know two sides and an angle that is not the included angle.

Say you know sides of length 9 and 7, and the angle opposite the side of length 7 is 30°. You want the third side.

Here, the Law of Cosines still works, but it gets awkward because the unknown side is tangled up in the cosine term. You'll either need to use the Law of Sines first to find an angle, then come back to the Law of Cosines, or solve a quadratic equation.

The Law of Sines goes:

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

So in our example: 7 / sin(30°) = 9 / sin(B). 643. That gives you sin(B) = 9 × sin(30°) / 7 ≈ 0.Then B ≈ 40° (or 140° — which is the ambiguous case*, more on that in a second).

Once you have the second angle, the third is just 180° minus the other two, and you can go back to the Law of Sines to find the missing side.

The Ambiguous Case (SSA Only)

Here's a real-world gotcha: with SSA, you can sometimes get two valid triangles from the same numbers. This is called the ambiguous case, and it happens when the angle you know is acute and the side opposite it is shorter than the other known side. Practical, not theoretical.

If your calculator gives you two possible values for the missing angle (one acute, one obtuse), and both fit the geometry, you genuinely have two triangles that match the given information. In most textbook problems, only one is "expected," but in real geometry, both are mathematically valid.

Common Mistakes People Make

Forgetting to identify the included angle correctly. The Law of Cosines needs the angle between* the two known sides. If you plug in the wrong angle, you'll get a number, but it'll be wrong.

Leaving the answer squared. c² = 49 doesn't mean the third side is 49. You have to take the square root. This sounds obvious. It still happens all the time.

Mixing up degrees and radians. If your calculator is in radian mode and you punch in cos(50), you're not getting cos(50°). Always check the mode before you start.

Trusting the Pythagorean theorem on non-right triangles. If a problem doesn't say "right triangle," don't assume it is one.

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Rounding too early. If you're working through multiple steps, keep a few extra decimal places until the end. Rounding intermediate values can shift your final answer more than you'd think.

Practical Tips That Actually Help

Draw the triangle first. Even if it's rough. Knowing which angle is between which sides is much easier when you can see them. I know it sounds like kindergarten advice, but a 10-second sketch saves a 10-minute mistake.

Label everything on the sketch. Write the side lengths near the sides and the angle measures inside. Match the formula's letters to your drawing. This is the single most effective habit for avoiding setup errors.

Estimate before you calculate. Think about what the answer should* roughly be. If your two known sides are 5 and 8 and the included angle is 60°, the third side has to be somewhere between 3 (very small angle) and 13 (very large angle). If your calculator spits out 25, something went wrong.

Use a calculator that shows your work. A scientific calculator with a history view lets you check what you actually entered. Or use a graphing calculator, or just write each step on paper. Mental math is great until it isn't.

For real-world problems, mind the units. If your sides are in meters, your answer is in meters. Sounds silly, but in a rush, people mix up cm and inches, or forget to convert at all.

Double-check with the Law of Sines. If you found the third side using the Law of Cosines, you can verify it by computing all three angles via the Law of Sines and confirming they add up to 180°. It's a built-in

error-checker that costs you nothing.

Memorize one form of the formula first. The Law of Cosines can be written in three equivalent forms (solving for a, b, or c). Master the one written in terms of c, and you can derive the others by symmetry. Don't try to memorize all three at once.

The Ambiguous Case: When Things Get Weird

Not every problem plays nicely. Sometimes you're given two sides and an angle that is not between them, known as the Side-Side-Angle (SSA) condition. This is the famous "ambiguous case" of the Law of Sines, and it can yield zero, one, or two valid triangles.

Here's the scenario: you know sides a and b, plus angle A (the angle opposite side a). You start by computing sin(B) using the Law of Sines:

sin(B) = b · sin(A) / a

Then you take the inverse sine to find B. But arcsin only gives you an angle between 0° and 90°. There might be a second solution where B' = 180° − B, giving a second valid triangle.

How to tell how many triangles exist:

If a < b · sin(A), no triangle is possible (the side is too short to reach).
If a = b · sin(A), exactly one right triangle.
If a ≥ b, exactly one triangle (the side is long enough that there's no ambiguity).
If b · sin(A) < a < b, you get two triangles.

This is where the "one triangle or two?" question comes from. Worth adding: if both values of B produce a valid angle C (meaning 180° − A − B is still positive), you have two solutions. If only one does, you have one.

Real-World Applications Worth Knowing

The Law of Cosines isn't just a classroom exercise. It shows up in surprising places:

Navigation and GPS. When a receiver can "see" two satellites, the distance to each is known, and the angle between the lines of sight can be computed. The Law of Cosines gives the distance between the satellites along the curved path.

Forestry and surveying. Measuring the distance across a canyon or river directly is often impossible. Surveyors measure two accessible sides and the angle between them, then apply the Law of Cosines to find the inaccessible distance.

Engineering and structural design. Trusses, bridges, and roof frames are built from triangles because triangles are rigid. The Law of Cosines helps engineers calculate forces and lengths in non-right-angled components.

Robotics and computer graphics. Inverse kinematics—figuring out joint angles to reach a target position—heavily relies on cosine relationships. Animation rigging uses it constantly.

Astronomy. Calculating the distance between celestial bodies, or the parallax angle of a star observed from different points in Earth's orbit, often reduces to a Law of Cosines setup.

Sports analytics. Advanced player tracking systems reconstruct 3D positions from multiple camera angles, and the math behind it is built on extensions of the same principles.

A Few Practice Problems to Cement the Idea

Work through these and you'll be solid:

A triangle has sides of length 7 and 10, with an included angle of 60°. Find the third side.
A triangle has sides of length 9 and 12, with an included angle of 120°. Find the third side.
A triangle has sides of length 5, 8, and 10. Find the angle opposite the side of length 10.4. A triangle has sides of length 4 and 11, with a non-included angle of 30° opposite the side of length 4. How many triangles are possible?
A surveyor stands 80 meters from a tree and 100 meters from a rock. The angle between these two lines of sight is 40°. How far apart are the tree and the rock?

Answers (worth checking only after you've worked them out):

≈ 8.66
≈ 18.44
≈ 95.7°
Zero triangles (4 < 11 · sin 30° = 5.5, so 4 < 5.5… wait, 11 · sin 30° = 5.5, and since 4 < 5.5, no triangle exists).
≈ 65.5 meters.

Wrapping It Up

The Law of Cosines is one of those rare tools that's both elegant and indispensable. In real terms, it takes a single formula and rescues you from every situation where the Pythagorean theorem taps out. Once you understand the structure—two sides, the angle between them, and you want the opposite side—it becomes almost automatic.

The real key is practice. Draw the triangle, label carefully, plug in slowly, and check your answer against reality. The first few times you apply it, you'll second-guess the formula and the angle. But that's normal. After a while, you'll spot cosine-law setups the way experienced drivers spot highway exits: without conscious effort.

Geometry is full of these moments where a single, well-placed idea unlocks a whole category of problems. The Law of Cosines is one of the biggest. Master it, and an entire branch of trigonometry opens up.