Ambiguous Triangle (SSA) Calculator
Ambiguous Triangle (SSA) Calculator is a mathematics calculator that helps you calculate triangle side lengths, angles, or area. The formula used is: h = b·sin(B); if a<h: 0 triangles; if a=h: 1 right triangle; if h<a<b: 2 triangles; if a≥b: 1 triangle. Ensure all inputs share the same unit system (e.g. all metres or all feet) unless the formula converts between units.
Formula
h = b·sin(B); if a<h: 0 triangles; if a=h: 1 right triangle; if h<a<b: 2 triangles; if a≥b: 1 triangle
Example Calculation
Inputs
- Side a (opposite to angle A): 6
- Side b (adjacent): 8
- Angle B (degrees): 40 °
With side a = 6, side b = 8 and angle B = 40°, the altitude is h = b·sin(B) = 8·sin(40°) ≈ 5.14. Because h < a < b (5.14 < 6 < 8), the data fits two different triangles, so the calculator returns both solutions for angle A and angle C.
Frequently asked questions
What is the ambiguous case (SSA)?
When you know two sides and a non-included angle, the side opposite the given angle can sometimes swing to two positions — producing zero, one, or two valid triangles. This calculator tells you how many exist and solves each one.
Why can SSA give two triangles?
If the altitude h = b·sin(B) is shorter than side a, but a is still shorter than side b, both an acute and an obtuse angle A satisfy the law of sines, so two distinct triangles are possible.
What units should I enter?
Enter the two sides in any single length unit (they must match) and angle B in degrees. Results follow the same units.
How accurate are the results?
Angles and sides are computed directly from the law of sines and rounded for display, so they are exact to the precision shown.
When does no triangle exist?
If side a is shorter than the altitude h = b·sin(B), the side cannot reach the baseline and no triangle can be formed.
How can I verify the calculation manually?
Use the displayed formula and work through the numbers step by step. If your manual result differs slightly, check rounding and unit conversions first.
Can I use this for planning and budgeting?
Yes. Run best-case, expected, and worst-case inputs to compare outcomes and make safer planning decisions.