SSA Triangle Calculator
Solve Side-Side-Angle triangles using the Law of Sines, detect ambiguous cases
Calculate SSA Triangle
Side opposite to the known angle
Second known side
Angle opposite to side a
Law of Sines Formula
a/sin(A) = b/sin(B) = c/sin(C)
Where a, b, c are sides and A, B, C are opposite angles
Example Calculation
Given: A = 46°, a = 31 cm, b = 27 cm
Step 1: Use Law of Sines to find angle B
sin(B) = (b × sin(A)) / a = (27 × sin(46°)) / 31
sin(B) = (27 × 0.719) / 31 = 0.626
Step 2: B = arcsin(0.626) = 38.8°
Step 3: C = 180° - A - B = 180° - 46° - 38.8° = 95.2°
Step 4: Find side c using Law of Sines
c = (a × sin(C)) / sin(A) = (31 × sin(95.2°)) / sin(46°) = 43.0 cm
SSA Triangle Cases
No Triangle
sin(B) > 1
No solution exists
Right Triangle
sin(B) = 1
One solution (B = 90°)
Ambiguous Case
0 < sin(B) < 1
Two possible solutions
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
This law relates the sides of a triangle to the sines of their opposite angles.
It's particularly useful for solving SSA (Side-Side-Angle) triangle problems.
Quick Tips
SSA is the ambiguous case of triangle solving
Two sides and an angle not between them
May have 0, 1, or 2 solutions
Check if sin(B) ≤ 1 for valid triangle
Understanding SSA Triangles
What is an SSA Triangle?
An SSA triangle is one where you know two sides and an angle that is not between those sides. This is also called the "ambiguous case" because it can result in zero, one, or two valid triangles.
Why is SSA Ambiguous?
- •The given side opposite the known angle may "swing" to create two different triangles
- •This happens when the known angle is acute and specific conditions are met
- •The Law of Sines helps determine if solutions exist
Solving Steps
Step 1: Apply Law of Sines
Calculate sin(B) = (b × sin(A)) / a
Step 2: Check Validity
If sin(B) > 1, no triangle exists
Step 3: Find Angle(s)
B₁ = arcsin(sin(B)), B₂ = 180° - B₁
Step 4: Complete Triangle(s)
Find remaining angles and sides
Important Notes
When No Triangle Exists:
- • The calculated sin(B) > 1
- • The given measurements are contradictory
- • Side 'a' is too short to reach the opposite vertex
When Two Triangles Exist:
- • sin(B) < 1 and both angles are valid
- • The angle A is acute
- • Side 'a' is shorter than side 'b'