Formula Used
A + B + C = 180°
a / sin A = b / sin B = c / sin C = 2R
c² = a² + b² − 2ab cos C
Area = √[s(s − a)(s − b)(s − c)]
b²m + c²n = a(d² + mn)
(BD/DC)(CE/EA)(AF/FB) = 1
(AF/FB)(BD/DC)(CE/EA) = −1 using directed lengths
OI² = R(R − 2r)
How to Use This Calculator
Choose a calculator mode first. Enter every requested measurement carefully. Select matching length and angle units.
Press Calculate to solve the problem. Review warnings before using rounded answers. Open detailed steps for theorem substitutions.
Enable diagrams for visual checking. Export results using CSV or PDF. Compare alternate methods when available.
Example Data
| Example | Known values | Main theorem | Expected observation |
|---|---|---|---|
| 3–4–5 triangle | a = 3, b = 4, c = 5 | Pythagorean converse | Right triangle |
| Equilateral | a = b = c = 8 | Equal-side theorem | Three 60° angles |
| SAS case | b = 7, c = 9, A = 48° | Law of Cosines | Unique triangle |
| SSA case | A = 35°, a = 8, b = 11 | Law of Sines | Possible ambiguous case |
| Coordinate case | A(0,0), B(6,0), C(2,5) | Distance and determinant | Centers and line equations |
Theorem Selection Guide
| Known information | Recommended method | Important note |
|---|---|---|
| Three sides | Law of Cosines and Heron’s formula | Check triangle inequality first |
| Two sides and included angle | Law of Cosines | This creates one triangle |
| Two angles and one side | Angle sum and Law of Sines | This creates one triangle |
| Two sides and non-included angle | Law of Sines | Zero, one, or two triangles may exist |
| Three coordinates | Distance, slope, and determinant formulas | Collinear points are invalid |
| Three cevian ratios | Ceva’s theorem | Product must equal one |
Frequently Asked Questions
What does SSS mean?
SSS means all three side lengths are known. The calculator checks triangle inequality first. It then applies cosine relationships.
Why can SSA produce two answers?
The sine function has supplementary angles. Both angles may satisfy the measurements. The calculator displays every valid triangle.
How are triangle centers calculated?
Centers use standard coordinate and barycentric formulas. Each center serves a different geometric role. Results include generated coordinates.
What is a degenerate triangle?
A degenerate triangle has zero area. One side equals the other two combined. Its vertices lie on one line.
How does Ceva’s theorem work?
Ceva tests three cevians for concurrency. Multiply the three directed segment ratios. A product of one confirms concurrency.
How does Menelaus’s theorem work?
Menelaus tests three points for collinearity. Directed ratios preserve important signs. The signed product must equal negative one.
Can the calculator use radians?
Yes, select radians before calculation. Input angles follow the selected unit. Results use that same angle unit.
Why do rounded results differ slightly?
Intermediate values retain extra precision. Displayed values use your precision selection. Small differences come from final rounding.
Can results be exported?
Yes, copy the complete result summary. CSV creates a structured data file. PDF opens a printable result document.