SSA Triangle Solver Calculator

SSA Triangle Input Form

Known angle A (degrees)

Angle A must be opposite side a.

Known side a

This side lies opposite angle A.

Known side b

This is the second known side.

Decimal precision

Length unit label

Optional label for lengths and area.

Example Data Table

Case	Angle A	Side a	Side b	Expected outcome
Two-solution case	35°	8	10	Two valid triangles appear.
Single-solution case	30°	5	10	One right triangle appears.
No-solution case	120°	7	8	No valid triangle exists.

Formula Used

1) Ambiguous SSA check

For the known angle A and side b, compute the altitude: h = b × sin(A). This helps identify whether the input creates zero, one, or two triangles.

2) Law of Sines

a / sin(A) = b / sin(B) = c / sin(C). First compute sin(B) = b × sin(A) / a, then evaluate all valid angle choices for B.

3) Remaining angle

C = 180° − A − B. A valid solution must keep angle C positive.

4) Missing side

c = a × sin(C) / sin(A). This gives the third side after angles are known.

5) Area and other measures

Area = ½ab sin(C), Perimeter = a + b + c, s = (a + b + c)/2, and r = Area / s.

How to Use This Calculator

Enter the known angle as A in degrees.
Enter side a, which must be opposite angle A.
Enter side b as the second known side.
Select the number of decimal places you want.
Optionally add a length unit such as cm, m, ft, or in.
Press Solve SSA Triangle.
Review the summary, case note, and all valid solutions shown above the form.
Use the CSV or PDF buttons to export the comparison table.

Frequently Asked Questions

1) What does SSA mean in a triangle problem?

SSA means you know two sides and one angle that is not included between those sides. This setup can create an ambiguous case with zero, one, or two valid triangles.

2) Why can the solver return two different triangles?

The Law of Sines can produce two possible values for one unknown angle. If both angle choices keep the triangle angle sum below 180°, then both triangles are valid.

3) When does no triangle exist?

No triangle exists when the known side opposite the given angle is too short. In acute cases, that happens when side a is smaller than the altitude h = b × sin(A).

4) What happens if angle A is obtuse?

If A is obtuse, side a must be the longest relevant side. When a is not longer than b, the input cannot form a valid triangle.

5) Does side a always have to oppose angle A?

Yes. This solver assumes standard triangle notation, where side a is opposite angle A, side b opposes angle B, and side c opposes angle C.

6) Can I use any length unit?

Yes. You can enter centimeters, meters, inches, feet, or any consistent length unit. The calculator treats the values numerically and displays the unit label you provide.

7) Why do my rounded values look slightly different?

The internal calculations use full precision, but the display uses your chosen decimal setting. Small visible differences can appear after rounding, especially in trigonometric calculations.

8) Is this solver suitable for homework and design checks?

Yes. It is useful for classroom work, quick verification, geometry practice, and early design checks. For critical engineering work, always validate with your required standards.