Law of Sines Calculator

Enter one side length and two angles of a triangle to find the third angle, the other two sides, and the triangle's area using the Law of Sines.

Quick Facts

Law of Sines
a/sin A = b/sin B = c/sin C
Each side length divided by the sine of its opposite angle is the same for every side of the triangle.
Angle sum
A + B + C = 180°
The three interior angles of any triangle always add up to 180 degrees.
Circumradius
a/sin A = 2R
The common ratio also equals twice the radius of the triangle's circumscribed circle.

Your Results

Calculated
Angle C
-
C = 180° − A − B
Side b
-
b = a·sin B / sin A
Side c
-
c = a·sin C / sin A
Triangle Area
-
Area = ½·b·c·sin A

Ready

Enter one side and two angles, then press Calculate.

Formula and Method for the Law of Sines

In any triangle, the ratio of a side's length to the sine of the angle opposite it is the same for all three sides: a/sin A = b/sin B = c/sin C, where a, b, and c are the side lengths and A, B, and C are the angles directly across from them. This common ratio also equals 2R, twice the radius of the triangle's circumscribed circle. This calculator solves the standard AAS/ASA case: given one side and two angles, it finds the third angle, the other two sides, and the triangle's area.

How the calculation works

Enter one side length (a) and the two angles you know: angle A, which must be the angle directly opposite side a, and a second angle B. The calculator first finds the third angle using the triangle angle-sum property, C = 180° − A − B. It then uses the Law of Sines ratio a/sin A to solve for the remaining sides: b = a·sin B / sin A and c = a·sin C / sin A. Finally, it computes the triangle's area from the two solved sides and the angle between them, Area = ½·b·c·sin A (side b and side c both meet at vertex A, so A is the included angle between them).

Common mistakes

  • Degrees vs. radians: this calculator expects angles in degrees, not radians.
  • Invalid angle sums: the two entered angles must add up to less than 180°, and each must be greater than 0° — otherwise no triangle exists.
  • Mismatched side/angle pairing: side a must be the side opposite angle A, not any other side. The Law of Sines only works when each side is paired with the angle directly across from it.
  • The ambiguous SSA case: if you actually know two sides and a non-included angle (rather than two angles and a side), more than one triangle can fit the data — that case is not what this AAS/ASA calculator solves.

Real-world applications

  • Surveying and navigation: find distances across obstacles (rivers, canyons, buildings) by measuring two angles from a known baseline length.
  • Engineering and construction: reconstruct triangular dimensions when only one length and two angles can be directly measured.
  • Astronomy and geodesy: triangulate distances to distant points using angles observed from two known locations.
  • Trigonometry education: check hand-solved AAS/ASA triangle problems and see intermediate steps.

Frequently Asked Questions

What is the Law of Sines formula?
The Law of Sines states that a/sin A = b/sin B = c/sin C for any triangle, where a, b, and c are side lengths and A, B, and C are the angles opposite those sides. This common ratio also equals 2R, twice the radius of the triangle's circumscribed circle.
When can I use the Law of Sines to solve a triangle?
The Law of Sines solves a triangle when you know two angles and any side (AAS or ASA), or two sides and a non-included angle (SSA, the ambiguous case). This calculator handles the AAS/ASA case: enter one side, its opposite angle, and a second angle.
What is the ambiguous case (SSA) of the Law of Sines?
When you know two sides and an angle that is not between them (SSA), the Law of Sines can produce zero, one, or two valid triangles, because the sine function gives the same value for an angle and its supplement. Always check whether each candidate solution has angles summing to less than 180 degrees.
How do I find a triangle's area after solving it with the Law of Sines?
Once you know two sides and the included angle between them, the area is Area = ½ × side1 × side2 × sin(included angle). This calculator computes it automatically as ½·b·c·sin A after solving for sides b and c.

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