Law Of Sines Calculator

What this calculator does

So, you have a triangle where a side and its opposite angle are paired up, and you want one missing piece. This tool uses the law of sines to find it. Working with two pairs, side A with angle A and side B with angle B, you give it any three of those four values and it returns the fourth.

A dropdown chooses what to solve for, an angle or a side, and the remaining three boxes are yours to fill.

How to use it

1. Choose what to calculate for: angle A, angle B, side A, or side B.
2. Enter the other three values.
3. Press Calculate.

The tool assumes you know each side alongside the angle across from it, which is the pairing the law of sines needs.

The law of sines

The law of sines says that in any triangle, each side divided by the sine of its opposite angle gives the same result. Written for two of the pairs:

side A / sin(angle A) = side B / sin(angle B)

That single equation ties four quantities together, so knowing any three lets you solve for the last. To find a side, the tool rearranges to multiply across; to find an angle, it isolates the sine and takes the inverse sine. Either way it is the same proportion, read in whichever direction the unknown demands.

Why sides pair with opposite angles

The pairing at the heart of the law is intuitive once you picture it. In any triangle, the bigger a side is, the bigger the angle facing it across the triangle, and the smaller the side, the smaller its opposite angle. The law of sines makes that loose observation exact: side and opposite angle do not just grow together, they grow in a fixed proportion, the same ratio for all three pairs in the triangle. That is why a side always travels with the angle across from it, never the angle next to it.

It works for any triangle

Unlike the Pythagorean theorem, which is confined to right triangles, the law of sines holds for every triangle, whether its angles are all acute, or one is a right angle, or one is obtuse. This is what makes it one of the main tools for solving general triangles. When you have a matched side and angle plus one more fact, it is often the quickest route to the missing measurement. One thing worth knowing is that solving for an angle from two sides and an angle can occasionally allow two valid triangles, a quirk of the geometry rather than the arithmetic; for straightforward pairings the answer is unique.

A worked example

Suppose angle A is 30 degrees, angle B is 45 degrees, and side B is 10, and you want side A. The law of sines says side A over sin 30 equals 10 over sin 45. Rearranged, side A equals sin 30 times 10 divided by sin 45, which is about 7.07. So the side opposite the 30 degree angle is roughly 7.07 units long.

Questions people ask

What does the law of sines let me find?

A missing side or angle in a triangle, as long as you have a side paired with its opposite angle and one more value from a second pair.

Which side goes with which angle?

Each side pairs with the angle opposite it, across the triangle, not the angle beside it. That opposite pairing is what the law relates.

Does it work for non-right triangles?

Yes. The law of sines holds for any triangle, acute, right, or obtuse, which is a large part of why it is so useful.

How does it solve for an angle?

It isolates the sine of the unknown angle in the proportion, then takes the inverse sine to recover the angle itself.

When would I use the law of cosines instead?

When your known values do not include a matched side and opposite angle, such as three sides or two sides with the angle between them. The triangle angle calculator draws on both laws.

References

On the law of sines. In any triangle, each side divided by the sine of its opposite angle gives the same ratio, which lets a missing side or angle be found.

1. Eric W. Weisstein, "Law of Sines," from MathWorld, a Wolfram resource, on the proportion between the sides of a triangle and the sines of their opposite angles.
2. Eric W. Weisstein, "Law of Cosines," from MathWorld, a Wolfram resource, on the companion law used when a matched side and opposite angle are not available.