Triangle Identities
Solve triangles — SSS, SAS, ASA, SSA. Laws of sines and cosines, area formulas.
Reference
Right triangle
- Pythagorean
- a² + b² = c² (hypotenuse c)
- sin θ
- = opposite / hypotenuse
- cos θ
- = adjacent / hypotenuse
- tan θ
- = opposite / adjacent
- Area
- = ½ · base · height
Any triangle
- Law of sines
- a / sin A = b / sin B = c / sin C = 2R (circumradius)
- Law of cosines
- c² = a² + b² − 2ab · cos C
- Area (SAS)
- = ½ · a · b · sin C
- Area (Heron)
- = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
- Sum of angles
- A + B + C = 180°
Solving strategies
| Given | Strategy |
|---|---|
| SSS (3 sides) | Law of cosines to find any angle; law of sines for others |
| SAS (2 sides + included angle) | Law of cosines for third side; law of sines for remaining angles |
| ASA / AAS | Sum of angles for third angle; law of sines for sides |
| SSA (ambiguous case) | Law of sines — may yield 0, 1, or 2 solutions |
Useful points
- Incenter
- Intersection of angle bisectors — inscribed circle
- Circumcenter
- Intersection of perpendicular bisectors — circumscribed circle
- Centroid
- Intersection of medians — center of mass
- Orthocenter
- Intersection of altitudes
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