Calculator inputs
Choose a mode, enter your known values, and submit to calculate the missing side, target angle, or the full triangle.
Example data table
| Case | Known values | Main result | Area | Perimeter |
|---|---|---|---|---|
| SAS example | a = 8, b = 11, C = 47° | c ≈ 8.0603 | ≈ 32.1796 | ≈ 27.0603 |
| SSS example | a = 7, b = 8, c = 9 | C ≈ 73.3985° | ≈ 26.8328 | 24 |
| Balanced SSS | a = 13, b = 14, c = 15 | A ≈ 53.1301° | 84 | 42 |
Formula used
For a missing side
c² = a² + b² − 2ab cos(C)
Use this when two sides and the included angle are known. After finding c, the remaining angles, area, and perimeter can be derived.
For a missing angle
cos(C) = (a² + b² − c²) / 2ab
Take the inverse cosine of the result to obtain the angle in degrees. The same pattern works for A and B.
Area relationship
Area = ½ab sin(C)
For SAS cases, this gives area directly. In SSS cases, the calculator also checks area with Heron’s formula.
Triangle validation
a + b > c, a + c > b, b + c > a
These conditions ensure the three sides can form a real triangle before angle and area calculations are attempted.
How to use this calculator
- Select a mode based on the values you already know.
- Enter sides for SSS, or two sides plus included angle for SAS.
- Pick the number of decimal places you want in the output.
- Submit the form to show results above the calculator.
- Review the table, graph, and optional step breakdown.
- Export the finished result set as CSV or PDF when needed.
Frequently asked questions
1) When should I use the law of cosines?
Use it when you know three sides, or when you know two sides and their included angle. It is especially useful when right-triangle shortcuts do not apply.
2) What is the difference between SAS and SSS?
SAS means two sides and the included angle are known. SSS means all three sides are known. The calculator supports both cases directly.
3) Why does the calculator reject some side combinations?
Some values cannot form a triangle. The calculator checks the triangle inequality first, because invalid side sets produce impossible angles and areas.
4) Does this tool return radians too?
Yes. The result table includes angles in both degrees and radians, which helps when switching between geometry, calculus, or programming workflows.
5) Can I use decimal values for side lengths?
Yes. Decimal entries are supported for all sides and angles, so the calculator works well for measurements from drawings, experiments, and engineering estimates.
6) What does the triangle type output mean?
The tool classifies the triangle by sides and by angles. You can see whether it is scalene, isosceles, or equilateral, and acute, right, or obtuse.
7) Is the graph drawn to scale?
Yes. The Plotly figure is built from the solved side lengths, so the displayed shape reflects the calculated geometry within normal screen rendering limits.
8) What can I export from this page?
You can export the result metrics as CSV or capture the result panel and plot as a PDF. This makes reporting and sharing easier.