Determine The Remaining Sides And Angles Of The Triangle Abc

You’re staring at a triangle labeled ABC. You know side a is 8 units, angle B is 40°, and side c is 10 units. The problem asks you to find everything else — the remaining sides and angles of the triangle ABC — and your brain immediately goes: Wait, which law do I use again?

Yeah. Been there.

It’s one of those geometry skills that feels abstract right up until you need it. Consider this: then it’s suddenly the only thing standing between you and a finished homework set, a working piece of code, or a correctly cut roof truss. Let’s walk through it like we’re sitting at a kitchen table with a stack of scratch paper.

What Does It Mean to Solve Triangle ABC?

"Determine the remaining sides and angles" is just the formal way of saying: you have a puzzle with three known pieces, and you need to find the other three.

Every triangle has six parts — three sides (usually lowercase a, b, c) and three angles (uppercase A, B, C). The standard convention is clean: side a sits opposite angle A, side b opposite B, side c opposite C. Also, if you stick to that labeling, the formulas behave. If you mix them up, you’re in for a world of pain.

To solve the triangle, you need any three parts, provided one of them is a side. (Three angles alone only gives you the shape — similar triangles — not the size.)

The possible starting combos have names:

ASA (Angle-Side-Angle)
AAS (Angle-Angle-Side)
SAS (Side-Angle-Side)
SSS (Side-Side-Side)
SSA (Side-Side-Angle) — the one everyone dreads

Each combo dictates your toolkit. That’s the first decision point.

Why This Skill Actually Matters

Look, I get it. And if you’re a student, it’s a grade. If you’re an engineer, surveyor, game developer, or navigator, it’s Tuesday.

GPS triangulation? Solving triangles. Calculating the height of a cell tower without climbing it? Solving triangles. Writing a ray-tracing engine for a video game? Plus, you’re solving thousands of triangles per frame. Even cutting crown molding for a corner that isn’t 90° — yep, triangle math.

The textbook makes it look like symbol pushing. In practice, it’s measurement without touching*. That’s the superpower.

How to Determine the Remaining Sides and Angles

The workflow is almost always the same: identify your knowns, pick your law, solve for one missing piece, then cascade.

### The Easy Ones: ASA and AAS (Law of Sines)

If you have two angles and a side — any side — you’re in Law of Sines territory.

First step: find the third angle. Also, triangle angles sum to 180°. That's why always. No exceptions in Euclidean geometry. Angle C = 180° – Angle A – Angle B.

Done. Now you have three angles and one side. The Law of Sines says:

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

Plug in your known side and its opposite angle. Also, that gives you the "common ratio" (the circumdiameter, if you want to get fancy). Then cross-multiply to get the other two sides.

Example: A = 30°, B = 70°, a = 12.* C = 80°.* 12 / sin(30°) = 24. That’s your ratio. b = 24 × sin(70°) ≈ 22.55.* c = 24 × sin(80°) ≈ 23.64.*

Boom. Solved.

### Two Sides and the Included Angle: SAS (Law of Cosines First)

Here you know a, b, and the angle C between them. You cannot use Law of Sines yet — you don’t have an angle-side opposite pair.

You need the Law of Cosines to find the third side c:

c² = a² + b² – 2ab cos(C)*

Take the square root. Now you have three sides and one angle. Now you can switch to Law of Sines to find a second angle (usually the smaller one, to avoid ambiguity — more on that in a sec). Then subtract from 180° for the third.

Watch your calculator mode. Degrees vs. radians is the silent killer of test scores. I’ve seen brilliant students get zero credit because they computed cos(60) in radian mode and got -0.95 instead of 0.5. Check the little "DEG" or "RAD" indicator every single time.

### Three Sides: SSS (Law of Cosines Only)

No angles given. Just a, b, c.

For more on this topic, read our article on how many days until feb 28 or check out what is the gcf of 24 and 36.

Use Law of Cosines to find the largest angle first — the one opposite the longest side. Day to day, why? Because the cosine function is one-to-one on 0° to 180°, but the sine function isn’t. If you use Law of Sines to find a large angle, you might accidentally get its supplement (the acute version) and not realize it.

Formula rearranged for angle C:

cos(C) = (a² + b² – c²) / 2ab*

Hit inverse cosine (arccos). Get C. Repeat for a second angle (or use Law of S

ines for the smaller one). Then find the third angle by subtraction.

Example: a = 7, b = 8, c = 10.* Largest side is c, so find C first.* cos(C) = (49 + 64 – 100) / (2 × 7 × 8) = 13 / 112 ≈ 0.116.* C ≈ arccos(0.116) ≈ 83.3°.* Now use Law of Sines for angle A (opposite smallest side, guaranteed acute):* sin(A) / 7 = sin(83.3°) / 10 → A ≈ 44.0°.* B = 180° – 83.3° – 44.0° = 52.7°.*

Triangle solved. No ambiguity, no drama.

### The Danger Zone: SSA (The Ambiguous Case)

Two sides and a non-included angle. This is where triangles get weird — or don’t exist at all.

Given a, b, and A (where a is opposite A). You’re hunting for angle B using Law of Sines:

sin(B) = (b × sin(A)) / a*

Three things can happen:

sin(B) > 1 → No triangle. Side a is too short to reach the baseline. The data describes an impossible shape.
sin(B) = 1 → One right triangle. B = 90°* exactly. Side a just kisses the baseline.
0 < sin(B) < 1 → Two possible angles. B₁ = arcsin(value)* (acute) and B₂ = 180° – B₁* (obtuse). Both might work. You must test each: if A + B₂ < 180°, you have two valid triangles. If A + B₂ ≥ 180°, only the acute B₁ works.

The "Swing" Mental Model: Imagine side a swinging from angle A like a pendulum. Side b is the ground.

Too short (a < b sin A*)? Misses the ground entirely.
Just right (a = b sin A*)? Touches at a perfect right angle.
Long but not too long (b sin A < a < b*)? Swings across the ground in two spots — one acute, one obtuse.
Long enough (a ≥ b*)? Only hits once, on the "far side." The obtuse swing goes backward, off the ray.

Always draw it. A thirty-second sketch saves ten minutes of algebraic confusion.

The Hidden Trap: Precision vs. Accuracy

You followed the steps. Calculator in degree mode. Laws applied correctly. Answer: B = 43.9998°.

Round at the end. Not in the middle.

If you round sin(70°)* to 0.Carry full calculator precision (or at least 5–6 decimals) through the chain. 56, then use that* rounded number to find the next angle… error compounds. 94, then multiply by 24 to get 22.Round only the final reported values — typically to the nearest tenth for angles, nearest hundredth for sides, unless the problem specifies otherwise.

And label your triangle consistently. Side a* is always opposite Angle A*. If you swap them in your diagram, the Laws of Sines and Cosines will happily give you the wrong answer for the wrong triangle. The math doesn’t know you mislabeled; it only knows the symbols you fed it.

Why This Still Matters

GPS doesn’t ask satellites for distances directly. It measures time delays — how long signals take to arrive. Also, those times become distances. Your position? On top of that, **Trilateration. The intersection of three spheres. And projected onto the ground? ** Triangle solving at planetary scale.

A structural engineer calculating loads on a truss bridge? Force vectors resolve into triangles. The roof pitch, the rafter length, the collar tie — every cut list comes from solving triangles.

The animator rigging a character’s elbow? Inverse kinematics. The software solves a triangle (upper arm, forearm, target wrist position) sixty times a second so the hand lands exactly on the doorknob.

You aren't learning "how to solve triangles.** That is the job. " You're learning **how to recover missing reality from partial measurements.The notation is just the interface.

Next time you see a triangle problem, don't ask "Which formula?What can I reach? " Ask: **"What do I know? What’s the path from here to there?

The laws are just the vehicle. The destination is the missing piece of the world.