SSA Triangle Calculator
Enter angle A, side a opposite angle A, and side b. The calculator checks the ambiguous case and returns every valid triangle.
Formula Used
Given: angle A, side a opposite A, and side b.
Law of sines: sin(B) / b = sin(A) / a
Angle B: B = sin-1((b × sin(A)) / a)
Second possible angle: B₂ = 180° - B
Third angle: C = 180° - A - B
Missing side: c = a × sin(C) / sin(A)
Area: Area = 1 / 2 × b × c × sin(A)
SSA height test: h = b × sin(A)
How to Use This Calculator
- Enter the known angle A.
- Select degrees or radians.
- Enter side a, which is opposite angle A.
- Enter side b, the second known side.
- Choose the decimal precision for your answer.
- Press the calculate button.
- Review whether the inputs form zero, one, or two triangles.
- Use the CSV or PDF button to save the result.
Example Data Table
| Angle A | Side a | Side b | Expected case | Reason |
|---|---|---|---|---|
| 30° | 7 | 10 | Two triangles | a is greater than h but less than b. |
| 30° | 5 | 10 | One triangle | a equals h, so B is 90°. |
| 30° | 4 | 10 | No triangle | a is smaller than h. |
| 110° | 15 | 8 | One triangle | An obtuse known angle allows only one valid triangle. |
SSA Triangle Guide
Understanding SSA Triangles
An SSA triangle gives one angle, the side opposite that angle, and another side. This data can be tricky. It may create no triangle, one triangle, or two different triangles. The reason is the sine rule. A sine value can match an acute angle and an obtuse angle. Both may fit the same known side data.
Why the Ambiguous Case Matters
The ambiguous case appears when the known angle is not between the two known sides. The calculator first checks the sine ratio for angle B. If that ratio is greater than one, no real triangle is possible. If it equals one, angle B is ninety degrees. If it is less than one, a second angle may also work.
What the Tool Calculates
This calculator solves every valid SSA result. It finds angle B, angle C, missing side c, perimeter, semiperimeter, area, height, circumradius, and inradius. It also classifies the triangle by side type and angle type. The output keeps alternate solutions separate, so comparison is simple.
Practical Use Cases
Students can use it to study trigonometry homework. Teachers can show how two triangles can share the same SSA inputs. Engineers, surveyors, designers, and builders can test field measurements before drawing layouts. The graph gives a quick visual check. The export buttons help save results for reports.
Accuracy Notes
SSA calculations depend on clean inputs. Use positive side lengths. Keep the known angle between zero and one hundred eighty degrees. Round only the final answer when possible. Very small rounding changes can make a borderline triangle appear possible or impossible. When measurements come from real work, add tolerances and check the result against the physical situation.
Reading the Results
Start with the solution count. Then compare angle totals. Each valid triangle should total one hundred eighty degrees. Review the missing side and area next. If two solutions appear, both satisfy the same SSA data. Choose the one that matches your drawing, direction, survey note, or problem statement.
Saving and Sharing Results
CSV files support spreadsheets. PDF reports are useful for notes. Keep the exported inputs with each answer. That makes checking easier and reduces mistakes during revision.
FAQs
1. What does SSA mean in triangles?
SSA means side, side, angle. It gives two side lengths and one angle that is not between those two sides.
2. Why can SSA produce two triangles?
The sine rule can return one acute angle and one obtuse angle. Both may satisfy the same side and angle data.
3. When is no SSA triangle possible?
No triangle is possible when the opposite side is too short to reach the other known side at the given angle.
4. What is side a in this calculator?
Side a is the side opposite angle A. This matching is required for the law of sines calculation.
5. What is side b?
Side b is the second known side. It is opposite the unknown angle B in standard triangle notation.
6. Can I use radians?
Yes. Select radians from the angle unit field. The calculator converts the angle internally before solving.
7. Why is the height test useful?
The height test compares a with b times sin(A). It quickly shows whether zero, one, or two triangles may exist.
8. Can I export the answers?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary report.