Triangle Solver
Solve any triangle given SSS, SAS, ASA, or AAS using law of sines and cosines.
Solve any triangle using SSS, SAS, ASA, or AAS
What Is the Triangle Solver?
The Triangle Solver uses the Law of Sines and Law of Cosines to find all missing sides, angles, and area of any triangle given three pieces of information. It supports SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), and AAS (two angles and non-included side) configurations.
Formula
How to Use
Select the mode (SSS, SAS, ASA, or AAS) from the dropdown. Enter the known values in the input fields that appear. Angles should be in degrees. Click Solve to compute all remaining sides and angles, plus the area and perimeter.
Example Calculation
SAS: a=7, b=10, C=60° c² = 7²+10²−2(7)(10)cos(60°) = 49+100−70 = 79 c = √79 ≈ 8.888 A = arcsin(7·sin(60°)/8.888) ≈ 42.95° B = 180°−60°−42.95° = 77.05° Area = (1/2)(7)(10)sin(60°) ≈ 30.31 sq units
Understanding Triangle
The Law of Sines and Law of Cosines are the two fundamental theorems for solving arbitrary triangles. Together, they cover every possible case: SSS, SAS, ASA, AAS, and (with care) SSA.
Triangles appear everywhere in engineering and science. Structural engineers analyze forces in triangulated frameworks (trusses). Surveyors use triangulation to measure distances and map terrain. Navigation uses the triangle formed by origin, destination, and intermediate waypoints.
The special case where C = 90° reduces the Law of Cosines to c² = a² + b², recovering the Pythagorean theorem. The formula Area = (1/2)ab·sin(C) shows that for a fixed perimeter, the maximum area is achieved by an equilateral triangle.
Frequently Asked Questions
What is the Law of Cosines used for?
The Law of Cosines generalizes the Pythagorean theorem to non-right triangles. Use it for SSS (finding all angles) and SAS (finding the side opposite the known angle).
What is the Law of Sines used for?
The Law of Sines relates sides and their opposite angles. Use it for ASA and AAS. Beware the "ambiguous case" (SSA): there may be 0, 1, or 2 valid triangles.
What is the ambiguous case (SSA)?
When given two sides and a non-included angle (SSA), there may be zero, one, or two possible triangles. This tool checks all possibilities and reports how many solutions exist.
What is the triangle inequality?
A valid triangle requires that the sum of any two sides exceeds the third side: a+b>c, a+c>b, b+c>a. Also, the sum of all angles must equal 180°.
How is area computed for non-right triangles?
Using SAS: Area = (1/2)ab·sin(C). Using SSS: Heron's formula. Using the base and height: Area = (1/2) × base × height.