How this instrument works
Angle-side-angle names one of the classical congruence conditions from Euclidean geometry: fix two angles and the side that joins them, and exactly one triangle satisfies those three measurements — Euclid works through the proof in Book I, Proposition 26 of the Elements. That length has to be the one caught between the two given angles for that proof to apply, which is why the field here is labelled Side c (between A and B) rather than left as a generic third measurement.
The remaining angle costs nothing extra to find: every planar triangle's three angles add to a flat 180°, so Angle C falls straight out as 180° minus the other two before any side length even enters the picture. What is left is a scale problem, not a shape problem — two fixed angles already pin down a triangle's shape, the way any two similar triangles share their corners regardless of size. Side c then fixes that scale, and the Law of Sines, written c ⁄ sinC = a ⁄ sinA = b ⁄ sinB, hands back the two missing lengths from that single shared ratio.
Compare that to angle-angle-side, where the known length sits outside the two given angles rather than between them: AAS reaches the same triangle, but only after first finding the third angle and then applying the Law of Sines to a length that was never the included one. Swap in a side-side-angle triangle instead and the certainty disappears altogether — two sides plus a non-included angle can swing shut on zero, one, or two valid triangles for the same three numbers. ASA never has that trapdoor, because fixing both angles first removes every degree of freedom the ambiguous case depends on.
- Type the two known angles into Angle A and Angle B, in degrees, then enter the length of the side between them into Side c (between A and B).
- Angle C (computed) fills in on its own as 180° minus your two entered angles — there is nothing to type there.
- Read Side a and Side b, the two lengths the Law of Sines recovers from the ratio sideC ⁄ sin(angleC).
- Keep Angle A plus Angle B under 180°; if the two entered angles already use up the full straight angle, no positive Angle C — and no triangle — remains.
Worked example — 45°, 60°, and a 10-unit side
Take a triangle built from Angle A = 45°, Angle B = 60°, and the included Side c = 10 units. Angle C falls out immediately, with nothing else needed: 180° − 45° − 60° = 75°, or 1.3089969389957472 radians once converted, and that is the only third angle any triangle with this A and B could ever have.
Side a and Side b still need the ratio. Side a = c·sinA ⁄ sinC = 10 × sin45° ⁄ sin75° = 10 × 0.7071067811865476 ⁄ 0.9659258262890683 ≈ 7.320508075688773. Side b = c·sinB ⁄ sinC = 10 × sin60° ⁄ sin75° = 10 × 0.8660254037844386 ⁄ 0.9659258262890683 ≈ 8.965754721680534 — and those two figures are the only pair of side lengths that close this particular triangle.
Questions
What does angle-side-angle actually guarantee?
A unique triangle. Euclid's Elements, Book I, Proposition 26, proves that two angles and the included side determine every remaining side and angle with no second possibility, which is why ASA is counted as a full congruence condition rather than merely a similarity rule that fixes shape but not size.
Why must the given side sit between the two given angles?
Because that is the exact configuration Euclid's proof is built around: the side anchors both angles at once, leaving no freedom for a second triangle to satisfy the same three numbers. A known side lying outside the two angles is a different case, angle-angle-side, which reaches an answer only after first computing the missing angle.
How is ASA different from the ambiguous SSA case?
Side-side-angle gives two sides and a non-included angle, and that combination can produce zero, one, or two valid triangles depending on whether a swinging side misses, just touches, or crosses the target vertex twice. ASA carries no such ambiguity: fixing two angles first locks the triangle's shape, so the Law of Sines returns exactly one pair of sides, every time.
Where does the Law of Sines ratio come from?
Drop a perpendicular from one vertex to the opposite side and it splits the triangle into two right triangles that share that height. Writing the shared height two different ways, once per right triangle, and setting the two expressions equal produces the constant ratio side ⁄ sin(opposite angle) — the identity behind every Law of Sines calculation.
What happens if Angle A plus Angle B reaches 180 degrees?
There is no triangle left to solve. A flat or negative Angle C means the two entered angles already exhaust the 180° available to a plane triangle, so the two angles must leave a strictly positive remainder before Side a and Side b mean anything at all.
Does the unit chosen for Side c matter?
Only in that it has to stay consistent. The Law of Sines is a pure ratio, so Side c in metres returns Side a and Side b in metres, and the same input in inches returns inches — mixing units partway through is the one way to turn a correct formula into a wrong answer.