How to Solve a Triangle – SSS, SAS, ASA/AAS & Right Triangles
How to Solve a Triangle – Choose the Right Method for the Values You Know
Triangle solving is primarily a method-selection problem. Once you classify what is known, the correct theorem usually becomes clear.
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What Does It Mean to Solve a Triangle?
To solve a triangle means finding its missing side lengths and angles from the measurements already known. A triangle has six basic measurements—three sides and three angles—but the method depends on the pattern of known information.
Which Triangle Formula Should You Use?
| Known information | Start with | Why |
|---|---|---|
| Right triangle + two sides | Pythagorean theorem | The 90° angle gives a²+b²=c². |
| Right triangle + side + acute angle | sin, cos or tan | Trig ratios directly connect sides and angles. |
| SSS | Law of Cosines | No angle is known. |
| SAS | Law of Cosines | Two sides and their included angle determine the third side. |
| ASA / AAS | Angle sum → Law of Sines | Two angles determine the third. |
| SSA | Law of Sines + ambiguity check | There may be 0, 1 or 2 valid triangles. |
Core Theorems
Worked Examples
Right triangle — legs 6 cm and 8 cm
Then tanA=6/8, so A≈36.87° and B≈53.13°. Check: 36.87+53.13+90=180°.
SAS — a=7, b=10, C=60°
c≈8.888
Now use the Law of Sines or Law of Cosines to find A and B.
ASA — A=50°, B=65°, a=12
Then b=12·sin65°/sin50°≈14.20 and c≈14.20.
SSS — sides 7, 8 and 9
C≈73.40°
Continue similarly for the other angles and verify the total is 180°.
The SSA Ambiguous Case
With two sides and a non-included angle, the Law of Sines may yield two possible angles because sinθ = sin(180°−θ). A correct solution must test the supplementary angle and verify whether each resulting triangle is valid.
How to Check Your Triangle
Practice Questions
Practice 1 — Legs 5 and 12
Answer: hypotenuse 13; acute angles ≈22.62° and 67.38°.
Practice 2 — Can 3, 4 and 8 form a triangle?
No. 3+4=7 is not greater than 8, so the triangle inequality fails.
Practice 3 — A=45°, B=75°, a=10
Answer path: C=60°. Then use the Law of Sines to find b and c.
Questions People Ask
Open any question for a detailed explanation, not just a one-line answer.
What information do I need to solve a triangle?
Usually three independent measurements, including at least one side. SSS, SAS, ASA and AAS normally determine a unique triangle. Three angles alone do not determine size. SSA requires an ambiguity check.
How do I know whether to use the Law of Sines or Cosines?
Use the Law of Cosines for SSS and SAS. Use the Law of Sines when you already know an opposite side-angle pair, especially ASA/AAS problems.
Can Pythagoras solve any triangle?
No. The Pythagorean theorem is only for right triangles. The Law of Cosines generalizes the same idea to non-right triangles.
Why do the angles add to 180 degrees?
In Euclidean geometry, the interior angles of every triangle sum to 180°. That lets you find the third angle whenever two are known and gives a powerful answer check.
What is the SSA ambiguous case?
SSA means two sides and a non-included angle. The sine of an angle equals the sine of its supplement, so the same data can sometimes produce two valid triangles.
How do I check a triangle answer?
Check the 180° angle sum, triangle inequality, side-angle ordering and any special relationship such as Pythagoras for a right triangle.
Can I solve a triangle with three angles only?
You can determine its shape but not its scale. At least one side length is required to fix the size.
Where can I solve all these cases automatically?
The Triangle Calculator supports multiple known-data cases. Quick Solve shows concise working and Learn Mode explains the theorem and reasoning in depth.
Related CalcTypes Resources
Methodology
Methods are organized by the pattern of known measurements. Examples retain enough precision to show the relationship and use validity checks such as the angle sum and triangle inequality.