Recover all three interior angles from three valid side lengths.
Triangle Geometry
Enter triangle data
Private calculation in your browser
1EnterProvide the known values
2CalculateResults update automatically
3VerifyReview the details and units
Try an example
Angles use degrees. Keep all lengths in one consistent unit and coordinates in one Cartesian system. Enter finite decimal or scientific-notation values; supported nonzero magnitudes are 1e-50 through 1e50. Inputs stay on this device.
Result
Enter valid values to see the result.
Your entries are calculated in this browser and are not submitted to 365CALCS.COM.
Triangle geometry connects side lengths, angles, centers, and special lines inside one three-sided figure.
Use it to identify which side, angle, center, or theorem relationship the entered measurements support.
The relationship
Write the rule before calculating
A=acos((b²+c²−a²)/(2bc)), with cyclic forms for B and C.
See the structure
What the calculation is doing
oppositeθhypotenuseA=acos((b²+c²−a²)/(2bc)), with cyclic forms for B and C.
Worked interpretation
Read the result in context
Sides 3, 4, and 5 give a right angle opposite side 5.
Interpret with care
Important boundary
A triangle must satisfy its inequality and its known measurements may allow no solution, one solution, or an ambiguous pair of solutions.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Quick guide
How to use this calculator
Choose the data pattern or theorem mode when the calculator offers one.
Enter only the side lengths, angles, ratios, or coordinates labelled by the active mode.
Read the main result first, then inspect the validated companion measurements and theorem check.
Calculation method
Apply the selected triangle relationship
A=acos((b²+c²−a²)/(2bc)), with cyclic forms for B and C.
The engine rejects degenerate triangles, impossible angle sums, failed strict inequalities, undefined ratios, and inconsistent theorem data before returning a result.
Worked example
Worked example
Sides 3, 4, and 5 give a right angle opposite side 5.
A=acos((b²+c²−a²)/(2bc)), with cyclic forms for B and C.
Supported inputs
Precision and limits
Units
Use one consistent length unit. Areas are reported in the square of that unit; angles are in degrees; coordinate tools use the entered Cartesian coordinate system.
Numeric range
Nonzero entered magnitudes must lie from 1e-50 through 1e50. Integer generators and exploratory modes also disclose tighter per-field limits.
Geometry
Ordinary triangle pages require a non-degenerate Euclidean triangle. The strict triangle inequalities and an interior-angle sum of 180° govern accepted data. Coordinate pages also reject vertex triples whose cross product is indistinguishable from zero at double-precision scale.
Precision
Calculations use finite double-precision arithmetic and normally display 12 significant digits. Coordinate-center tools also report offsets from vertex A; use those offsets when a very large coordinate translation rounds distinct absolute centers to the same displayed coordinate. The Pythagorean page retains a simple exact integer radical when the squared data make that representation available.
