How the law of sines works
Every triangle carries a hidden constant. Take any side, divide it by the sine of the angle sitting opposite that side, and note the answer. Do it again with a different side and its opposite angle. You get the same number. Do it a third time and you get it again.
That shared value is the whole engine. It means a single matched pair, one side together with the angle facing it, gives you a ratio you can carry to any other corner of the triangle. Know one pair and one more piece, and the rest of the shape unfolds. It is a far lighter tool than the law of cosines, with no squares and no rearranging, which is why it is worth reaching for first whenever it applies.
The formula
Sides a, b and c each sit opposite angles A, B and C. Opposite is the important word: the angle must be the one looking across the triangle at that side, not touching it. To find a missing side you cross-multiply one pair against another:
And to find a missing angle, flip it the other way: sin B = b sin A ÷ a. That second form is the one that hides a trapdoor, and the rest of this page is largely about it.
Worked example
Two angles and a side. A surveyor stands at one end of a 12 metre baseline. From that end a landmark sits 40 degrees off the line; from the far end it sits 75 degrees off. How far is the landmark from each end?
First, the free angle: C = 180° − 40° − 75° = 65°.
b = 12 × sin 75° ÷ sin 40° = 12 × 0.9659 ÷ 0.6428 =
18.03 metres
c = 12 × sin 65° ÷ sin 40° = 12 × 0.9063 ÷ 0.6428 =
16.92 metres
The triangle closes with an area of 98.06 square metres. Notice that the two angles alone were never enough: they fix the triangle's shape but not its size, and infinitely many triangles share those angles. The 12 metre baseline is what pinned down which one.
The ambiguous case, where one measurement makes two triangles
Here is the trapdoor. Suppose you know two sides and an angle, but the angle is not the one between them. You measure side a as 7, side b as 9, and angle A as 40 degrees. Reasonable numbers. One triangle, surely?
Two. And they are not subtly different:
| First triangle | Second triangle | |
|---|---|---|
| Angle B | 55.73° | 124.27° |
| Angle C | 84.27° | 15.73° |
| Side c | 10.84 | 2.95 |
| Area | 31.34 | 8.54 |
One of these triangles is nearly four times longer along side c than the other, and covers almost four times the area. Both are completely valid. Both use your three measurements exactly. If you measured a real field and wrote down those numbers, you genuinely cannot tell from the numbers alone which field you walked.
The cause is simple once you see it. You solve for angle B by taking an inverse sine, and sine cannot distinguish an angle from its supplement: sin 55.73° and sin 124.27° are the same value to the last decimal place. Your calculator picks the acute one because it has to pick something, and the obtuse one sits there quietly being equally true.
How to tell how many triangles you have
You do not have to guess. Drop a perpendicular from the top corner down to the base and call its height h = b sin A. That height is the shortest possible reach from that corner to the base. Now compare your side a against it:
- a < h: your side is too short to reach the base at all. No triangle exists.
- a = h: it reaches by exactly touching, perpendicular. One right triangle.
- h < a < b: it is long enough to reach, and it can swing to touch on either side of the perpendicular. Two triangles.
- a ≥ b: long enough that one of those two landing spots falls outside the triangle entirely. One triangle.
In the example above, h = 9 sin 40° = 5.79, and 5.79 < 7 < 9, which lands squarely in the two-triangle band. The calculator runs this test on every SSA problem and tells you which case you are in before it hands over any answers.
Worth saying plainly: none of this is a flaw in anyone's working, and it is not a trick question either. It is a real geometric fact about what three measurements can and cannot pin down, and it is the reason careful surveyors record a fourth measurement they do not strictly need. If you know in advance that your triangle is obtuse, or you can measure any third side, the ambiguity vanishes immediately. The law of cosines has no ambiguous case at all, because inverse cosine returns obtuse angles directly. And if you would rather hand over whatever you happen to know and let something else choose the method, the triangle calculator does exactly that.