Every triangle has six elements: three sides (a, b, c) and three angles (A, B, C), where capital letters denote the angle opposite the lowercase side. To completely "solve" a triangle means to find all six values. The remarkable result is that you only need three pieces of information—as long as at least one of them is a side—to determine everything else. The two main tools for doing this are the Law of Sines and the Law of Cosines.
The Law of Sines
The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C). This ratio—a side divided by the sine of its opposite angle—is the same for all three side-angle pairs in any triangle. It follows that if you know any two angles and one side, you can find the remaining sides.
Use the Law of Sines when you have: AAS (two angles and the side opposite one of them) or ASA (two angles and the included side between them). Since the three angles of a triangle always sum to 180°, knowing two angles immediately gives you the third.
Worked Example (AAS)
Given: angle A = 35°, angle B = 62°, side a = 8. Find sides b and c.
- Find C: C = 180° − 35° − 62° = 83°.
- Apply the Law of Sines: 8/sin(35°) = b/sin(62°) = c/sin(83°).
- Solve for b: b = 8 × sin(62°)/sin(35°) = 8 × 0.8829/0.5736 ≈ 12.31.
- Solve for c: c = 8 × sin(83°)/sin(35°) = 8 × 0.9926/0.5736 ≈ 13.84.
The Law of Cosines
The Law of Cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C). Notice that when C = 90°, cos(C) = 0, and the formula reduces to c² = a² + b²—the Pythagorean theorem. Use the Law of Cosines when you have: SSS (all three sides) or SAS (two sides and the angle between them).
Worked Example (SAS)
Given: a = 5, b = 7, C = 48°. Find side c and angles A and B.
- Find c: c² = 5² + 7² − 2(5)(7)·cos(48°) = 25 + 49 − 70 × 0.6691 = 74 − 46.84 = 27.16. So c = √27.16 ≈ 5.21.
- Find A using Law of Sines: sin(A)/5 = sin(48°)/5.21. sin(A) = 5 × 0.7431/5.21 ≈ 0.7130. A = arcsin(0.7130) ≈ 45.5°.
- Find B: B = 180° − 48° − 45.5° = 86.5°.
For right triangles specifically, the Right Triangle Calculator uses the Pythagorean theorem directly and is the simplest tool for that special case.
The Ambiguous SSA Case: When There Are 0, 1, or 2 Solutions
The most confusing scenario in triangle solving is SSA (two sides and an angle opposite one of them—but not the included angle). This is called the ambiguous case because, depending on the values, there may be zero, one, or two valid triangles.
Given side a, side b, and angle A (the angle opposite a): compute h = b·sin(A), the altitude from the vertex of angle B.
- If a < h: side a is too short to reach the base. No triangle exists.
- If a = h: side a exactly reaches the base at a right angle. Exactly one right triangle exists.
- If h < a < b: side a can swing to either side, creating two distinct triangles.
- If a ≥ b: only one triangle is possible, since the longer side leaves no room for an alternate configuration.
When two triangles are possible, solve for angle B using sin(B) = b·sin(A)/a, then check both B₁ = arcsin(result) and B₂ = 180° − B₁. Both values yield valid triangles as long as B₂ + A < 180°. Use our Triangle Calculator to handle all cases automatically.
Area Formulas
Once you know two sides and an included angle, the area is: Area = ½ab·sin(C). This is the most general triangle area formula.
If you know all three sides but no angles, use Heron's formula. First compute the semi-perimeter s = (a + b + c)/2. Then: Area = √[s(s−a)(s−b)(s−c)]. For example, a triangle with sides 6, 8, and 10 (a right triangle): s = 12, Area = √[12 × 6 × 4 × 2] = √576 = 24. Check: ½ × 6 × 8 = 24. ✓
Common Mistakes and How to Check Your Answer
- Using degrees vs. radians: Most triangle problems use degrees. Make sure your calculator is in degree mode before computing sines and cosines.
- Forgetting the ambiguous case: Whenever you have SSA, always check whether a second triangle is possible before declaring a unique solution.
- Not checking the angle sum: After solving, verify that A + B + C = 180°. Any deviation (beyond small rounding errors) indicates a mistake.
- Inverting the Law of Sines: sin(A)/a = sin(B)/b is the same ratio, but it's easy to accidentally write a/sin(A) = sin(B)/b when cross-multiplying. Double-check the setup.
The Pythagorean Theorem Calculator handles the 90° special case, but for all other triangles, the Law of Sines and Law of Cosines between them cover every possible configuration. Master the rule for choosing which law to apply—based on the given information—and any triangle becomes solvable.