Audited 05 Aug 2026·Last updated 28 Oct 2026·3 citations·Tier 2·0 uses

Ssa Triangle Calculator

Resolve the ambiguous SSA case: from side a, its opposite angle A and side b, find whether zero, one or two triangles exist and each possible angle B.

Ssa Triangle Calculator

degrees
Number of valid SSA triangles
2
How many distinct triangles the measurements describe: 1 or 2. Inputs that form no triangle produce an error instead.
First possible angle B
36.8699degrees
Supplementary angle B
143.1301degrees

Background.

Give someone two sides of a triangle and the angle between them, and there is only one triangle they can draw. Move the angle so it sits opposite one of the known sides instead, and the certainty disappears. That side-side-angle arrangement is why geometry courses accept SAS, ASA, AAS and SSS as proofs that two triangles are congruent but never SSA: the same three measurements can fit two different triangles, one, or none at all. The one exception, hypotenuse-leg, is SSA with the angle fixed at 90°.

Picture it physically. Draw angle A at a corner and lay side b along one of its arms, ending at a point C. Now hinge a rod of length a at C and swing it down towards the other arm. Too short, and it never reaches. Exactly as long as the height of C above that arm, and it touches once, at a right angle. Longer than the height but shorter than b, and it crosses the arm in two places, giving two triangles. At least as long as b, and only one crossing lands on the right side of the corner.

The page settles which case you are in with the law of sines. Since a ÷ sin A = b ÷ sin B, the sine of the unknown angle B is b × sin A ÷ a. A sine above 1 means no triangle exists. Otherwise two angles share that sine, B and 180° − B, and each is kept only if it leaves room for a third angle, meaning A + B stays below 180°.

What comes back is the number of valid triangles and both candidate values of B. The rest of each triangle, angle C and side c, follows from the angle sum and the law of sines again but is not calculated here.

What is ssa triangle calculator?

An SSA triangle is one specified by two sides and an angle that is not between them: here side a, the angle A opposite it, and a second side b that meets side c at A. Because an angle and its supplement have the same sine, the law of sines can yield two candidate values for the angle B opposite side b, and the measurements may describe zero, one or two genuine triangles. This situation is known as the ambiguous case.

How to use this calculator.

  1. Label the triangle so the known angle is A and the side facing it is a. The other known side, b, is one of the two sides that meet at A.
  2. Enter angle A in degrees, greater than 0 and less than 180. The two sides can be in any length unit, as long as both use the same one.
  3. Read the number of triangles first. When it is 2, both B values describe real triangles; when it is 1, only the first B is valid and the supplementary value is shown for reference.
  4. To finish a triangle, take C = 180° − A − B and c = a × sin C ÷ sin A, repeating with the second B if there are two solutions.

The formula.

sin(B)=b*sin(A)/a; test both B=arcsin(value) and 180 degrees-B
a · the known side opposite angle AA · the known angle, in degreesb · the second known side, adjacent to angle AB · the angle opposite side b: arcsin(b × sin A ÷ a) or its supplement 180° − B

The law of sines states a ÷ sin A = b ÷ sin B, so sin B = b × sin A ÷ a. The product b × sin A is the perpendicular height of the far end of side b above the other arm of angle A, which means the test sin B ≤ 1 is the same question as whether side a is long enough to reach that arm. If it passes, the first candidate is B₁ = arcsin(b × sin A ÷ a), between 0° and 90°, and the second is B₂ = 180° − B₁. B₁ counts when A + B₁ < 180°; B₂ counts when A + B₂ < 180° and B₂ differs from B₁. If A is 90° or more, B₂ is at least 90° and always fails, so an obtuse or right A allows one triangle at most. The count is most sensitive near the boundaries, where a tiny change in a, b or A can turn two triangles into one or none.

A worked example.

Example

With a = 10, A = 30° and b = 12, the law of sines gives sin B = 12 × sin 30° ÷ 10 = 12 × 0.5 ÷ 10 = 0.6. The first candidate is B = arcsin(0.6) = 36.8699°, the smaller acute angle of a 3-4-5 right triangle. Its supplement is 180° − 36.8699° = 143.1301°. Both pass the angle-sum test, since 30° + 36.87° = 66.87° and 30° + 143.13° = 173.13° are each under 180°, so the page reports 2 triangles. They are very different shapes. The first has C ≈ 113.13° and c ≈ 18.39; the second has C ≈ 6.87° and c ≈ 2.39, a long thin sliver. Swap the sides to a = 12 and b = 10 and sin B falls to about 0.4167; the supplementary 155.38° no longer fits beside 30°, and only one triangle remains.

known Side B12
known Angle A30
known Side A10

Frequently asked questions.

Why isn't SSA a congruence rule when SAS is?
Because SSA measurements do not fix the shape. In the page's own example, sides 10 and 12 with a 30° angle opposite the 10 fit one triangle with a third side near 18.39 and another with a third side near 2.39. Two triangles that share the measurements but differ this much cannot be guaranteed congruent.
When there are two triangles, which one is correct?
Both are valid geometrically, so the choice needs information the three measurements do not contain: whether angle B is acute or obtuse, a rough length for side c, or a sketch of the real object. Without it, report both solutions rather than picking the first.
What changes when angle A is obtuse?
An obtuse angle leaves no room for a second obtuse angle, so the supplementary B always fails and there is at most one triangle. That triangle exists only when side a is longer than side b; otherwise a cannot reach across to close the shape.
Why does the page show a second angle B when it says there is only one triangle?
The supplementary value is always displayed so you can see why it was rejected. When the count is 1, adding that angle to A gives 180° or more, leaving nothing for angle C, so it does not describe a real triangle.
What happens when side a exactly equals b × sin A?
That is the boundary between no triangle and two, for example a = 6, A = 30° and b = 12. The circle of radius a just touches the other arm of angle A, so there is exactly one triangle, a right triangle with B = 90°, and the page reports one solution with B at 90°. Make a even slightly shorter and no triangle exists; make it slightly longer and two appear.

References& sources.

  1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
  2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
  3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.

How this page was produced

Published by
Quanta Calculator
Overseen by
Daniel, Quanta Calculator Researcher
Primary sources
3 cited below
Method
sin(B)=b*sin(A)/a; test both B=arcsin(value) and 180 degrees-B
Published
Last verified

Built with AI assistance and verified by automated tests against the cited sources — every worked example on this page is computed by the same code that runs the calculator. How we build and check calculators.

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