Law of Sines Calculator
Solve a triangle from two angles and one side.
Results update as you type.
How it works
The law of sines relates the sides of any triangle to the sines of their opposite angles, and it is the standard tool for solving triangles that are not right-angled. This law of sines calculator takes one angle, the side directly opposite it, and a second angle, then finds the remaining two sides and the third angle. The rule states that each side divided by the sine of its opposite angle gives the same ratio throughout the triangle, so once you know one side and its opposite angle you have that ratio, and you can multiply it by the sine of any other angle to get the side across from it. The third angle is found immediately from the fact that all three interior angles sum to one hundred eighty degrees. Surveyors use the law of sines to find distances that cannot be measured directly, navigators use it to fix positions, and engineers and physicists use it whenever forces or structures form oblique triangles. It applies to the angle-angle-side and angle-side-angle cases, which always have a single well-defined solution, making it a dependable companion to the law of cosines for the remaining triangle-solving situations.
a / sin A = b / sin B = c / sin C
The law of sines states that in any triangle the ratio of each side to the sine of its opposite angle is the same constant. Given one angle and the side opposite it, that ratio is fixed, so any other side is found by multiplying the ratio by the sine of the angle opposite it. The third angle follows because the interior angles of a triangle always add up to one hundred eighty degrees.
Source:Trigonometry (law of sines)
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter angle a (°), side a (opposite a) and angle b (°).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Side b (opposite B) updates instantly as you type — no button to press.
Frequently asked questions
What is the law of sines?
The law of sines states that in any triangle, the ratio of a side's length to the sine of the angle opposite it is constant across all three sides. Written as a over sine A equals b over sine B equals c over sine C, it links sides and angles so that knowing some of them lets you find the rest. It works for all triangles, not just right-angled ones.
When can I use the law of sines?
Use it when you know two angles and any one side, or two sides and an angle opposite one of them. The angle-angle-side and angle-side-angle cases, which this calculator handles, always give a single clear solution. The side-side-angle case can sometimes produce two possible triangles, known as the ambiguous case, and needs extra care.
How do I find the third angle?
Since the three interior angles of any triangle always add up to exactly one hundred eighty degrees, subtract the two known angles from one hundred eighty to get the third. The calculator does this first, then uses the law of sines to compute the sides opposite each angle. This is why supplying two angles is enough to pin down the whole triangle's shape.
What is the difference from the law of cosines?
The law of sines pairs each side with its opposite angle and works best when you know angle-side pairs. The law of cosines relates all three sides to one angle and is the right choice when you know two sides and the included angle, or all three sides. Together they cover every case of solving an oblique triangle.
Does it work for right triangles?
Yes. The law of sines applies to every triangle, including right-angled ones, where the sine of the ninety-degree angle equals one. For right triangles the simpler ratios of basic trigonometry, sine, cosine, and tangent, are usually quicker, but the law of sines still gives correct results. It is most valuable precisely when no right angle is present.