First, name the information you have
Use lower-case letters a, b and c for side lengths and capital A, B and C for the opposite angles. This opposite pairing matters: side a is always across from angle A. Many wrong answers begin with a side matched to the wrong angle.
A general triangle needs three independent measurements, including at least one side. Three angles alone fix the shape but not its size; a small and large triangle can have the same three angles. The common input patterns are named by the order of known sides (S) and angles (A).
| Known information | Typical first method | What to watch |
|---|---|---|
| SSS — three sides | Cosine rule for one angle | The two shorter sides must add to more than the longest. |
| SAS — two sides and included angle | Cosine rule for the opposite side | The known angle must be between the known sides. |
| ASA or AAS — two angles and one side | Angle sum, then sine rule | Match every side with its opposite angle. |
| SSA — two sides and a non-included angle | Sine rule | There may be zero, one or two valid triangles. |
| Right triangle | Pythagoras or SOH-CAH-TOA | The hypotenuse is opposite 90° and is longest. |
The three relationships used most often
The interior angles always add to 180 degrees. If two angles are known, the third is immediate.
A + B + C = 180°The sine rule connects each side with the sine of its opposite angle. It is especially useful when you know an opposite side-angle pair.
a / sin A = b / sin B = c / sin CThe cosine rule is Pythagoras extended to any triangle. Use it with SSS or SAS.
c² = a² + b² − 2ab cos CSSS: three sides
Suppose a = 5, b = 7 and c = 8. Use the cosine rule rearranged for angle C:
cos C = (a² + b² − c²) / (2ab) = (25 + 49 − 64) / 70 = 1/7Therefore C ≈ 81.79°. Apply the same rearrangement for A, or use the sine rule after one angle is known. The result is approximately A = 38.21°, B = 60.00° and C = 81.79°. As a check, the angles total 180°.
Before calculating, test the triangle inequality. For every triangle, each pair of sides must add to more than the remaining side. A set 2, 3, 6 cannot close because 2 + 3 is less than 6.
SAS: two sides and the included angle
Worked example: sides 8 and 6 with included angle 60°
Let a = 8, b = 6 and C = 60°. The opposite side c comes from the cosine rule.
c² = 8² + 6² − 2(8)(6)cos60° = 64 + 36 − 48 = 52c = √52 ≈ 7.211Now the triangle is SSS. Use the cosine rule or sine rule for another angle, then subtract from 180° for the last. The included-angle detail is critical: if the 60° angle is not between the 8 and 6 sides, the input pattern is SSA instead.
ASA and AAS: two angles and one side
If A = 50°, B = 60° and a = 10, then C = 70°. An opposite pair is available—side a and angle A—so the sine rule gives the other sides.
b = a sin B / sin A = 10 sin60° / sin50° ≈ 11.305c = a sin C / sin A = 10 sin70° / sin50° ≈ 12.267ASA and AAS follow the same calculation sequence. The labels only describe whether the known side lies between the two known angles.
SSA and the ambiguous case
SSA is different because the sine rule may produce two angles with the same sine. For example, sin 30° equals sin 150°. After calculating an angle with inverse sine, the supplementary angle may also fit the supplied sides.
Take A = 30°, a = 7 and b = 10. The sine rule gives sin B = b sin A / a = 5/7. One answer is B ≈ 45.58°; the other is 180° − 45.58° = 134.42°. Both leave a positive third angle, so both triangles are valid. A good solver must report both rather than silently selecting one.
If the computed sine is greater than 1, no triangle exists. If the supplementary angle makes A + B reach or exceed 180°, only the first solution is valid.
Right triangles
With a 90° angle, use the right triangle calculator or work directly with Pythagoras and trigonometric ratios. For legs 3 and 4, the hypotenuse is √(3² + 4²) = 5. An acute angle can be found from tan A = opposite / adjacent.
Area as a useful check
When two sides and their included angle are known, area is half their product times the sine of that angle: K = ab sin C / 2. With three sides, use Heron’s formula. Let s = (a + b + c)/2, then K = √[s(s − a)(s − b)(s − c)]. For the 5–7–8 triangle, s = 10 and K = √(10·5·3·2) = √300 ≈ 17.321.
Common mistakes
- Matching an angle with an adjacent side instead of its opposite side in the sine rule.
- Entering degrees into a calculation set to radians, or converting twice.
- Using the cosine rule as though the known angle were included when it is not.
- Ignoring the second SSA solution.
- Rounding an intermediate angle or side too early. Keep full precision until the final display.
- Accepting a result without checking the 180° sum and whether the drawn proportions look reasonable.
The solver shows the selected method, all sides and angles, area and other derived properties. Calculation precision and validation limits are documented in Methodology & Accuracy.