You're staring at a triangle. The third? Maybe you're building a deck, cutting pipe, or helping a kid with homework. Two sides are known. Missing. The question is always the same: *how do I find the missing length?
Here's the thing — there isn't one answer. This leads to there are several. And which one you need depends entirely on what else you know.
What You're Actually Looking For
Determining the third side of a triangle means using the information you have — side lengths, angles, or both — to calculate the one measurement you don't have. Sounds simple. In practice, it's where most people get stuck because they reach for the wrong tool That's the part that actually makes a difference..
The three scenarios that cover 95% of real problems
Two sides and the angle between them — this is the Law of Cosines territory. Most versatile. Works on any triangle.
Two angles and one side — Law of Sines. Elegant, but you need to be careful about the ambiguous case.
Two sides and a non-included angle — also Law of Sines, but this is where the ambiguous case lives. Two possible triangles. One solution. Or none at all Less friction, more output..
Right triangle, two sides known — Pythagorean theorem. The one everyone remembers from school. But it only works when you have a 90° angle.
That's the map. Now let's walk each path Easy to understand, harder to ignore..
Why This Matters More Than You Think
Bad triangle math costs money. Real money.
A carpenter cuts a rafter short because they used Pythagoras on a non-right triangle. That's why a surveyor miscalculates a property line because they assumed the ambiguous case had only one answer. A programmer builds a collision detection system that fails when triangles get skinny — because floating-point precision hates near-degenerate triangles.
I've seen all three. The last one took me three days to debug.
But it's not just professionals. Ever tried to figure out if a TV fits in a corner? That's a triangle problem. Hanging a picture level on a sloped ceiling? Triangle. But setting up a tent with guy lines? You guessed it.
The math isn't hard. Knowing which math to use — that's the skill.
How to Actually Find the Missing Side
Right triangles: the Pythagorean theorem
You know this one. That's why a² + b² = c². The hypotenuse c is opposite the right angle. The legs are a and b.
Finding the hypotenuse: square both legs, add them, take the square root.
Finding a leg: square the hypotenuse, subtract the square of the known leg, take the square root.
Example: legs are 3 and 4. In real terms, hypotenuse = √(9 + 16) = √25 = 5. Classic 3-4-5 triangle. Works every time.
But — and this is crucial — only works on right triangles. Practically speaking, " Exactly 90°. Not "almost right.Plus, " Not "looks square. If you're not sure, don't use it.
Special right triangles: shortcuts worth memorizing
45-45-90 triangle: legs are equal. Hypotenuse = leg × √2.
30-60-90 triangle: short leg = x, long leg = x√3, hypotenuse = 2x.
These appear constantly in construction, design, and standardized tests. Practically speaking, memorize them. You'll use them more than you expect The details matter here..
Any triangle, two sides and the included angle: Law of Cosines
This is the workhorse. The formula:
c² = a² + b² − 2ab cos(C)
Where C is the angle between sides a and b, and c is the side opposite angle C.
Notice something? If C = 90°, cos(90°) = 0. Now, the last term vanishes. Think about it: you get c² = a² + b². But pythagoras falls out as a special case. That's not a coincidence — it's geometry telling you something deep.
How to use it:
- Label your triangle. Side a opposite angle A, side b opposite B, side c opposite C.
- Plug in the two known sides and the angle between them.
- Solve for the unknown side.
Example: a = 5, b = 7, angle C = 60°.
And c² = 25 + 49 − 2(5)(7)cos(60°)
c² = 74 − 70(0. 5)
c² = 74 − 35 = 39
*c = √39 ≈ 6.
Done. Works on obtuse triangles, acute triangles, any triangle.
Any triangle, two angles and a side: Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
This is cleaner when you have angles. But you need at least one complete pair — a side and its opposite angle Easy to understand, harder to ignore..
Typical workflow:
- Find the third angle (angles sum to 180°).
- Set up the ratio with your known side-angle pair.
- Solve for the unknown side.
Example: A = 40°, B = 70°, side a = 10. Even so, find side b. First: C = 180° − 40° − 70° = 70°.
So then: a/sin(A) = b/sin(B) → 10/sin(40°) = b/sin(70°)
*b = 10 × sin(70°) / sin(40°) ≈ 10 × 0. Think about it: 9397 / 0. 6428 ≈ 14.
Clean. Fast. But watch the next section.
The ambiguous case (SSA): two sides, non-included angle
It's where
