Law of Sines
The law of sines states that a/sin A = b/sin B = c/sin C for any triangle. Given two angles and the side opposite one of them, this solver finds the third angle by the 180° rule and then the two remaining sides.
One ratio, used twice
In any triangle, a longer side faces a bigger angle. The law of sines makes that vague ordering exact: each side divided by the sine of the angle facing it gives the same number, for all three pairs. One constant, shared by the whole triangle.
This solver uses it for the AAS case. Given two angles and a side opposite one of them, it finds the third angle from the angle sum and then both remaining sides from the shared ratio.
The structure is unusually simple. Two angles determine the third immediately, so all three are known before any side is computed. The known side and its opposite angle fix the common ratio, and each remaining side is that ratio times the sine of its own angle.
That ratio is reported on its own line rather than being folded into the results. It is the quantity both remaining sides are built from, so checking it once verifies both — and it has a geometric meaning of its own, being the diameter of the circle through all three vertices.
How to use this calculator
- Enter the two known angles in degrees Both positive, and together summing to less than 180 so that something is left for the third. A pair that sums to 180 or more describes no triangle and is rejected.
- Enter the side opposite the first angle This must be the side facing angle A, not either of the others. It sets the scale of the whole triangle — the shape is fixed by the angles alone.
- Read the third angle Taken from the 180-degree sum. It is computed before anything else, since both remaining sides need their own opposite angles.
- Check the ratio line, then the two sides The common ratio is shown explicitly. Both reported sides are that number times a sine, so a mistake in it shows up in both at once.
The formula, and where it comes from
a / sin A = b / sin B = c / sin C C = 180° − A − B b = (a / sin A)·sin B
The three equal quotients are the law itself. Any two can be paired into an equation, and each pairing solves for one unknown given three knowns — which is what makes the rule cover several configurations.
Here the pairing is anchored on the known side and its angle, so one quotient serves both unknowns. Computing it once and multiplying twice is fewer operations than two separate proportions, and keeps the results consistent by construction.
The angle sum is not part of the law of sines but is what makes the AAS case solvable at all. Without it only one of the two unknown sides would have a known opposite angle, and the second could not be reached.
Because the shape of a triangle is determined by its angles alone, the single given side does nothing but set the scale. Doubling it doubles both computed sides and leaves every angle untouched.
What each input means
- A First angle — form field “Angle A (°)”
- In degrees, strictly positive. The side you supply must be the one facing this angle. Units: degrees.
- B Second angle — form field “Angle B (°)”
- Also in degrees and positive. Together with A it must sum to less than 180, leaving room for the third angle. Units: degrees.
- a Side opposite A — form field “Side a (opposite A)”
- Must be positive. It carries the units of the answer and sets the overall size, since the angles have already fixed the shape.
- a / sin A Common ratio
- Not entered — computed from the given pair and reused for both unknown sides. It equals the diameter of the triangle's circumscribed circle.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A standard AAS triangle
Angles of 40 and 75 degrees with a side of 10 opposite the first. The third angle follows immediately, and both remaining sides come from one ratio.
Inputs Angle A (°) = 40, Angle B (°) = 75, Side a (opposite A) = 10
- Given A = 40°, B = 75°, side a = 10
- Third angle C = 180° − A − B = 65°
- Law of sines a / sin A = 10 / 0.642788 = 15.5572
- Side b b = (a/sin A)·sin B = 15.0271
- Side c c = (a/sin A)·sin C = 14.0996
Result C = 65°, b = 15.0271, c = 14.0996
The third angle is 65 degrees, so the angles rank 40, 65, 75 and the sides must rank the same way. The given side of 10 faces the smallest angle and is therefore the shortest of the three — both computed sides come out longer.
The common ratio is a little over 15, noticeably larger than any side. That is expected: it is the circumdiameter, and every side is a chord of that circle rather than a diameter of it.
A long, narrow triangle
Angles of 20 and 30 degrees with a side of 6 opposite the first. Both known angles are small, so the third is very large and the triangle is stretched.
Inputs Angle A (°) = 20, Angle B (°) = 30, Side a (opposite A) = 6
- Given A = 20°, B = 30°, side a = 6
- Third angle C = 180° − A − B = 130°
- Law of sines a / sin A = 6 / 0.34202 = 17.5428
- Side b b = (a/sin A)·sin B = 8.77141
- Side c c = (a/sin A)·sin C = 13.4386
Result C = 130°, b = 8.77141, c = 13.4386
The third angle is 130 degrees, and the side facing it dominates the other two. When one angle approaches a straight line the triangle flattens, and the sine of that angle falls back towards zero even as the angle grows.
The sine of 130 degrees equals the sine of 50 degrees, which is worth noticing: sine cannot distinguish an angle from its supplement. That is harmless here because the third angle was found by subtraction rather than by an inverse sine.
Reading the result
Sides and angles rank together
The largest side always faces the largest angle and the smallest faces the smallest. Comparing the two orderings is a complete check on a solved triangle, and it costs nothing.
The ratio is the circumdiameter
The shared quotient equals the diameter of the unique circle passing through all three vertices. That is why it exceeds every side: each side is a chord of that circle, and no chord is longer than the diameter.
Angles fix shape, one side fixes size
Two triangles with the same angles are similar, differing only by a scale factor. The given side supplies that factor, which is why every reported length is directly proportional to it.
When you would use this
Surveying an inaccessible point
Measuring one baseline and sighting the angles to a distant object from each end gives two angles and the side between the observation points. The remaining distances follow directly.
Completing a triangle in a construction
Where a design fixes two angles and one edge, the other two edges are determined. This computes them in one step, without a constructed perpendicular.
Assumptions and limitations
What this calculator assumes
- The triangle is planar and all angles are given in degrees.
- Both given angles are positive and sum to less than 180 degrees.
- The given side is opposite the first of the two given angles.
- The given side is strictly positive.
Where it stops being the right tool
- The ambiguous SSA case is deliberately excluded: two sides and a non-included angle can describe two triangles, one, or none, and this tool does not resolve that.
- The SAS and SSS configurations need the law of cosines instead, since neither offers a matched angle-and-opposite-side pair to start from.
- Degrees only, with no radian input or output.
Common mistakes
Supplying a side that is not opposite angle A
Why it happens. All three sides are equally easy to measure, and the field asks only for a length. Using the wrong one produces a triangle of the right shape but the wrong size, with no error to signal it.
How to avoid it. Trace from the vertex of angle A across to the side facing it. If the length you know sits somewhere else, relabel the triangle so that it does — the letters are yours to assign.
Attempting the ambiguous case here
Why it happens. Two sides and an angle feels like enough information, and the law of sines does apply to it. It just does not determine a unique answer, so a single reported triangle would be misleading.
How to avoid it. Recognise SSA when you have it. If the known angle is not between the known sides, expect zero, one or two solutions, and solve it as a separate problem.
Taking an inverse sine to find an unknown angle
Why it happens. It is the obvious rearrangement, but sine gives the same value for an angle and its supplement, so the inverse cannot tell an acute answer from an obtuse one.
How to avoid it. Find remaining angles from the 180-degree sum wherever possible, as this solver does. Subtraction has no ambiguity to resolve.
Key terms
Frequently asked questions
What is the AAS case?
Angle-Angle-Side: two angles and a side that is not between them. The third angle follows from the 180-degree sum, and once all three angles are known the single side scales the whole triangle.
Does this handle the ambiguous SSA case?
No. Two sides with a non-included angle can correspond to two different triangles, exactly one, or none at all, depending on the numbers. Resolving that needs a case analysis this tool does not perform.
Which side should I enter?
The one opposite the first angle you entered. That pairing is what fixes the common ratio; a length taken from elsewhere in the triangle gives a correctly shaped answer at entirely the wrong scale.
What is the ratio line showing?
The value shared by all three side-to-sine quotients. Both unknown sides are found by multiplying it by the sine of their own opposite angle, and geometrically it is the diameter of the circle through all three vertices.