Finding Angles of a Triangle- Methods and Formulas
What You Need to Know About Triangle Angles
A triangle has three interior angles. Period. Those angles always add up to 180°, and that's the foundation for every method you're about to learn.
That's it. That's the core fact. Every technique for finding missing angles boils down to this simple relationship or its variations.
The Angle Sum Theorem
This is your starting point. If you know two angles, you find the third by subtraction:
Third Angle = 180° − (Angle 1 + Angle 2)
Example: If you have 50° and 60°, the missing angle is 180° − 110° = 70°
This works for every triangle. No exceptions.
Finding Angles When You Know Side Lengths
The Angle Sum Theorem only gets you so far. When you're given side lengths, you need trigonometry.
The Law of Sines
Use this when you know:
- Two angles and one side, or
- Two sides and an angle opposite one of them
The formula is straightforward:
a/sin(A) = b/sin(B) = c/sin(C)
Where a, b, c are sides and A, B, C are their opposite angles.
The Law of Cosines
Use this when you know:
- All three sides, or
- Two sides and the included angle
Pick the angle you want to find, then solve:
c² = a² + b² − 2ab·cos(C)
Rearrange to isolate cos(C), then use inverse cosine to get the angle.
Special Triangle Rules
Some triangles have predictable angle patterns. Save yourself calculation time.
- Equilateral triangle: All angles are 60°
- Right triangle: One angle is always 90°. The other two add to 90°
- Isosceles triangle: Two angles are equal (the angles opposite the equal sides)
Comparison: Which Method to Use
| What You Know | Best Method |
|---|---|
| Two angles | Angle Sum Theorem |
| Two angles + one side | Law of Sines |
| Two sides + included angle | Law of Cosines |
| All three sides | Law of Cosines |
| Right triangle + one acute angle | SohCahToa (Trig ratios) |
How To: Finding a Missing Angle (Step-by-Step)
Here's the practical approach:
Step 1: Identify what you know
List the given angles and/or side lengths. Check if any angles are 90°.
Step 2: Choose your weapon
No angles given? → Law of Cosines. One angle given with two sides? → Law of Sines or Law of Cosines. Right triangle? → Trigonometric ratios.
Step 3: Set up your equation
Write the formula. Plug in your known values.
Step 4: Solve
Use your calculator. Make sure it's in degree mode, not radians, unless specified otherwise.
Step 5: Check your work
Add all three angles. They must equal 180°. If they don't, you made an error.
Common Mistakes to Avoid
- Calculator in wrong mode — Radians vs degrees will give you garbage results
- Using Law of Sines with SSA — This creates the ambiguous case with two possible solutions
- Rounding too early — Keep full precision until the final answer
- Forgetting the Angle Sum check — It's your built-in error detector
When You Have a Right Triangle
Right triangles are simpler. Use SohCahToa:
- Sin = Opposite / Hypotenuse
- Cos = Adjacent / Hypotenuse
- Tan = Opposite / Adjacent
Pick the ratio containing your known sides, set up the equation, and solve for the angle using inverse trig functions.
The Bottom Line
Finding triangle angles comes down to three tools: the Angle Sum Theorem for easy cases, and the Laws of Sines and Cosines for everything else. Know when to use each. Practice the setup. Check your answers.
That's all you need.