SSA Triangle Calculator
Solve SSA (side-side-angle) triangles instantly using the law of sines. Our free online ambiguous case calculator finds 0, 1, or 2 solutions for any triangle problem.
About This Calculator
Our SSA Triangle Calculator helps you solve side-side-angle triangles using the law of sines. This calculator handles the famous ambiguous case of triangle solving, where given two sides and a non-included angle can result in zero, one, or two valid triangles. Whether you are a student learning trigonometry, a teacher preparing geometry problems, an engineer working on structural analysis, or a professional needing quick triangle solutions, this tool provides instant and accurate results.
The calculator uses the law of sines formula: a / sin(A) = b / sin(B) = c / sin(C). Given sides a and b and non-included angle A, the calculator first computes the height h = b x sin(A). It determines the number of possible triangles by comparing side a against h and b: no triangle if a < h, one right triangle if a = h, two triangles if h < a < b and angle A is acute, and one triangle in all other cases. For each valid solution, the calculator computes angle B using the inverse sine function, finds angle C as 180 deg - A - B, and determines side c using the law of sines.
The ambiguous case of the law of sines is a classic topic in trigonometry and geometry education. Understanding the SSA condition is essential for mastering triangle congruence, as SSA does not guarantee a unique triangle like SSS, SAS, ASA, or AAS do. This calculator makes it easy to visualize and verify the possible triangle configurations for any set of SSA inputs. Simply enter your known values and instantly see whether a unique triangle, no triangle, or two possible triangles satisfy your measurements.
Frequently Asked Questions
What is an SSA triangle?
An SSA (side-side-angle) triangle is a triangle where you know two sides and a non-included angle. This is also called the ambiguous case because the given information may correspond to zero, one, or two distinct triangles.
How does the law of sines apply to SSA triangles?
The law of sines states that a/sin(A) = b/sin(B) = c/sin(C). For SSA triangles, you know sides a and b and angle A. Using a/sin(A) = b/sin(B), you can solve for sin(B). If sin(B) > 1, no triangle exists. If sin(B) = 1, one right triangle exists. Otherwise, two possible angles B (acute and obtuse) may yield valid triangles.
How many solutions can an SSA triangle have?
An SSA triangle can have 0, 1, or 2 solutions. Zero solutions occur when side a is shorter than the height from angle A. One solution occurs when the triangle is right-angled, when angle A is obtuse and a > b, or when the given sides form a single valid triangle. Two solutions occur when angle A is acute and side a is between the height and side b.
Can SSA prove triangle congruence?
No, SSA alone cannot prove triangle congruence. Unlike SSS, SAS, ASA, or AAS, the SSA configuration does not guarantee a unique triangle. This is why it is called the ambiguous case -- you must check whether zero, one, or two triangles satisfy the given measurements.
What is the height h in the SSA ambiguous case?
The height h is computed as h = b x sin(A), where b is the known side and A is the known angle. If side a is shorter than h, no triangle can exist. If a equals h, exactly one right triangle exists. If a is between h and b, two triangles exist. If a is greater than or equal to b, one triangle exists.
Is this SSA triangle calculator free to use?
Yes, this SSA triangle calculator is completely free to use with no registration required. All calculators on Calculy are free and accessible online for students, teachers, engineers, and anyone needing quick triangle solutions.
What formulas does the SSA triangle calculator use?
This calculator uses the law of sines: a/sin(A) = b/sin(B) = c/sin(C). Given sides a, b and angle A, it computes angle B using sin(B) = b x sin(A) / a. Angle C is found from C = 180 deg - A - B, and side c from c = a x sin(C) / sin(A). The calculator also checks the ambiguous case conditions using the height h = b x sin(A).
What is the difference between SSA and SAS triangles?
In SSA, you know two sides and a non-included angle, which leads to the ambiguous case with potentially 0, 1, or 2 solutions. In SAS, you know two sides and the included angle, which always gives a unique triangle solvable using the law of cosines. SAS is not ambiguous, while SSA requires careful analysis.