Law of Sines Calculator - Solve Triangles (AAS, ASA, SSA)
Use the law of sines to find unknown sides and angles in any triangle. Supports AAS, ASA, and SSA including the ambiguous case.
Select the configuration that matches your known values, enter them below, and get all sides, angles, and triangle properties instantly.
Law of Sines Calculator - Solve Triangles (AAS, ASA, SSA)
Use the law of sines to find unknown sides and angles in any triangle. Supports AAS, ASA, and SSA including the ambiguous case.
Law of Sines Examples
Four examples covering AAS, ASA, and both SSA cases.
| Configuration & Given Values | Missing Values | Notes |
|---|---|---|
| AAS: A=45°, B=60°, a=10 | C=75°, b≈12.25, c≈13.66 | C = 180−105 = 75°. b = 10⋅sin(60°)/sin(45°) ≈ 12.25. c = 10⋅sin(75°)/sin(45°) ≈ 13.66. |
| ASA: A=30°, c=12, B=50° | C=100°, a≈6.09, b≈9.33 | C = 180−80 = 100°. a = 12⋅sin(30°)/sin(100°) ≈ 6.09. b = 12⋅sin(50°)/sin(100°) ≈ 9.33. |
| SSA: a=15, b=10, A=60° | One solution: B≈35.26° | sin(B) = 10⋅sin(60°)/15 ≈ 0.5774. Only one B < 180−A is valid. |
| SSA: a=8, b=10, A=40° | Two solutions: B≈52.47° or B≈127.53° | Ambiguous case: sin(B) = 10⋅sin(40°)/8 ≈ 0.8035. Both arcsin values give valid triangles. |
About the Law of Sines Calculator
The law of sines is one of the two fundamental theorems for solving triangles (the other being the law of cosines). For any triangle with sides a, b, c and opposite angles A, B, C, the law states: a/sin(A) = b/sin(B) = c/sin(C). This common ratio is equal to the diameter of the triangle’s circumscribed circle (circumcircle), a beautiful geometric property.
The law of sines is applicable when you know at least one side and the angle opposite it, along with additional information that uniquely (or ambiguously) determines the triangle. Three configurations are supported by this calculator.
AAS (Angle-Angle-Side): You know two angles and a non-included side. Since the three angles of a triangle sum to 180°, the third angle is computed immediately as C = 180° − A − B. Then the remaining sides are found using b = a⋅sin(B)/sin(A) and c = a⋅sin(C)/sin(A). The solution is always unique.
ASA (Angle-Side-Angle): You know two angles and the side between them. The approach is similar to AAS: compute the third angle, then apply the sine rule to find the other two sides. The solution is again unique.
SSA (Side-Side-Angle): You know two sides and a non-included angle. This is the “ambiguous case.” Depending on the values, there may be zero, one, or two valid triangles. The calculator detects all cases: if the given angle is obtuse and the opposite side is longer than the adjacent side, there is exactly one solution; if the angle is acute, there may be two solutions if the opposite side is shorter than the adjacent side but long enough to reach the base line. The calculator reports both solutions when they exist.
The law of sines has broad applications in navigation, surveying, and engineering. Triangulation, the technique of determining a point’s location from angles measured at known reference points, relies on repeated application of the sine rule. In navigation, bearing and distance calculations between waypoints use trigonometric laws. In structural analysis, forces on truss members are resolved through sine-rule calculations when the geometry is defined by angles and one known side.
This calculator automates all three configurations, handles the ambiguous SSA case transparently, and reports all triangle properties: all three sides, all three angles, and the triangle type.
How to Use the Law of Sines Calculator
- Select the configuration that matches your known values: AAS (two angles and a non-included side), ASA (two angles and the included side), or SSA (two sides and a non-included angle).
- Enter the known values in the corresponding fields. Angles are in degrees.
- Click Calculate. The calculator applies the sine rule to find all unknown sides and angles.
- For SSA, check whether one or two solutions are reported. The ambiguous case is handled automatically.
- Click Reset to clear all fields and solve a new triangle.
Frequently Asked Questions
What is the law of sines?
The law of sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This common ratio equals the diameter of the circumscribed circle. It is used to solve triangles where at least one side-angle pair (side and its opposite angle) is known.
What is the ambiguous SSA case?
The SSA case (two sides and a non-included angle) is called ambiguous because it can produce zero, one, or two valid triangles. When the angle is acute and the opposite side is between the height of the triangle and the adjacent side, there are two possible triangles with different configurations. The calculator identifies both solutions automatically.
When should I use the law of sines versus the law of cosines?
Use the law of sines for AAS, ASA, and SSA configurations. Use the law of cosines for SAS (two sides and the included angle) and SSS (three sides known). The law of cosines avoids the ambiguity of the SSA case by solving a quadratic equation, whereas the law of sines uses a simpler ratio but must handle two possible arcsin values.
How do I enter angles in this calculator?
All angles should be entered in degrees. The calculator converts to radians internally for the trigonometric functions. Make sure that for AAS and ASA, the two entered angles sum to less than 180° so that the third angle is positive. For SSA, the entered angle must also be between 0 and 180 degrees.
What does the “triangle type” mean?
The calculator classifies triangles by their angles and sides. By angle: acute (all angles < 90°), right (one angle = 90°), or obtuse (one angle > 90°). By side: equilateral (all sides equal), isosceles (two sides equal), or scalene (no sides equal). These labels appear in the result section when a valid solution is found.
Can the law of sines be used for right triangles?
Yes. For a right triangle with the right angle at C, sin(C) = sin(90°) = 1, so the sine rule simplifies to a/sin(A) = b/sin(B) = c. This is consistent with the basic right-triangle trigonometry formulas sin(A) = a/c and sin(B) = b/c. The law of sines works for all triangles including right triangles.