About the Law of Sines Calculator
Given two angles and one side of a triangle, the law of sines lets you find everything else about it without ever needing a right angle. Our Law of Sines Calculator solves for the remaining side lengths and third angle from that starting information.
How It Works
You enter two known angles and the side length opposite one of them. The calculator finds the third angle by subtracting from 180°, then uses the constant ratio between each side and the sine of its opposite angle to find the two remaining sides.
Formula & Methodology
The law of sines states that within any triangle, the ratio of a side's length to the sine of its opposite angle is the same constant for all three sides. Once you know one such ratio (from the given side and its opposite angle), you can find any other side by multiplying that same ratio by the sine of its own opposite angle. The third angle comes first, though, from the simple fact that a triangle's interior angles always sum to 180°: subtract the two given angles from 180° to find the third. This calculator specifically asks for two angles and a side (AAS/ASA) rather than two sides and a non-included angle (SSA), because the SSA case can be ambiguous, sometimes producing zero, one, or two valid triangles, while ASA and AAS always produce exactly one.
Step-by-Step: Calculating It By Hand
- 1Subtract the two given angles from 180° to find the third angle.
- 2Divide the known side by the sine of its opposite angle to get the shared ratio.
- 3Multiply that ratio by the sine of the second angle to find its opposite side.
- 4Multiply that same ratio by the sine of the third angle to find the last side.
Examples
Standard triangle
With angle A = 50°, angle B = 60°, and side a = 10: angle C is 70°, side b is about 11.28, and side c is about 12.26.
Near-right triangle
With angle A = 45°, angle B = 89°, and side a = 7: angle C is 46°, and side b comes out close to 9.9, nearly the hypotenuse.
Advantages
- Solves for every remaining side and angle from just two angles and one side
- Avoids the ambiguous SSA case by design, always producing exactly one triangle
- Validates that the given angles actually form a legitimate triangle
- Faster and more reliable than working through the ratio by hand
Common Mistakes
- Confusing which side is opposite which angle when setting up the ratio
- Applying the law of sines to an SSA (side-side-angle) situation, where results can be ambiguous
- Forgetting to first find the third angle before working out the remaining sides
- Entering angles that sum to 180° or more, which can't form a valid triangle
Edge Cases to Watch For
- The two given angles must be positive and sum to less than 180°, or no valid triangle exists.
- This calculator sidesteps the 'ambiguous case' (SSA) entirely by requiring two angles instead, which always has a single solution.
- A very small angle produces a very small ratio contribution to its opposite side, and vice versa for large angles.
- As one angle approaches 180°, the triangle becomes increasingly degenerate (flat).
Common Use Cases
- Trigonometry coursework on oblique (non-right) triangles
- Navigation and surveying problems involving angles measured from two known points
- Astronomy and physics problems involving triangulated angles
- Verifying triangle solutions found by hand using the ASA or AAS method