Live case strip and SSA twin sheets - so an ambiguous given set never pretends to finish only one triangle.
Planar geometry sketch in degrees. SSA ≠ unique in the ambiguous window. Not a CAD or surveying instrument.
By Ahmet C. Toplutaş·Site owner & editor · Math tools
Last updated May 21, 2026 · Reviewed May 21, 2026 · Planning estimates only - not tax, medical, or investment advice
01
Congruence shortcuts like SSS and SAS earn their reputation because the given set usually finishes one triangle. SSA - two sides and an angle opposite one of them - is the exception students meet too late: the same numbers can describe zero, one, or two triangles.
Treating every numeric output as “the” triangle is how homework diagrams and calculator printouts silently diverge. One twin may match the sketch’s obtuse swing; the other is still mathematically faithful to the same SSA list.
This page’s framing is SSA ≠ unique. The panel names the solving case from your filled fields, completes the missing measures, and - when the ambiguous window opens - shows both twins side by side so “I got an answer” is not mistaken for “there is only one answer.”
Surveyors, CAD techs, and geometry students hit the same trap in different clothing: three numbers feel like a complete brief. Without naming the congruence case, those three numbers may under-specify (SSA twins) or over-specify (four fields fighting each other). Naming the case first is the literacy skill; the trig is secondary.
Decision framing: “Which given set did I enter, and how many triangles finish it?” - not “which formula can I force these six boxes into?”
02
Side a sits opposite angle A, b opposite B, c opposite C. Mixing that pairing is the fastest way to turn a correct law-of-sines step into a wrong figure. Lengths may use any consistent unit; angles are in degrees on this sketch.
Leave unknowns blank. The solver reads which three independent pieces you provided and lights the matching case on the case strip. Extra filled fields that overdetermine the figure are not a free “check” - clear what you are not given.
If your worksheet labels sides differently (sometimes informal notes reverse lowercase opposite capitals), remap before typing. The panel will happily solve the wrong labeled problem with perfect internal consistency.
03
SSS - three sides. Unique up to congruence when the triangle inequality holds. SAS - two sides and the included angle. ASA - two angles and the included side. AAS - two angles and a non-included side (third angle is 180° minus the sum). SSA - the ambiguous case family.
The case strip is the unique “what did I actually give?” affordance - different from a percent job switcher or an energy activity ladder. Congruence names are the pedagogy.
ASA versus AAS is easy to confuse when two angles and a side are present: ask whether the known side sits between the known angles. The strip’s label follows that geometric question, not alphabetical luck.
04
SSS and SAS lean on the law of cosines to recover an angle or the third side, then fill remaining angles from 180°. ASA/AAS and SSA lean on the law of sines: a/sin A = b/sin B = c/sin C. Floating-point noise can make an angle sum drift by a hair; the panel uses standard complementary closure after the primary trig step.
When sin B comes back greater than 1, there is no real angle - that is a “no triangle” signal, not a calculator failure. When arcsin returns an acute B, remember the supplementary angle 180°−B may also be a candidate in SSA before you discard it.
05
With acute angle A opposite side a, and side b known, compare a to the height h = b sin A. If a < h, no triangle. If a = h, one right triangle. If a ≥ b, one triangle. If h < a < b, two triangles - the classic ambiguous case. Obtuse or right A has its own zero-or-one rules.
When two solutions exist, the panel shows two measure sheets. Your diagram’s swing of the “loose” side picks the twin; the calculator cannot see the sketch on your paper. That is why SSA ≠ unique is a framing, not a bug report.
Teachers often hide the second solution by drawing only one figure. Good practice is to ask whether another triangle could share the same SSA list before you box a single area.
Height h = b sin A is the gatekeeper: it is the altitude from the vertex of A down to side b’s line when you swing side a. Visualizing that swing explains why “a just past h” produces two hits on the base line and “a shorter than h” produces none. Memorizing the inequality table without the picture is how students mix up a = h (one right triangle) with a ≥ b (one triangle of a different kind).
Try the SSA ambiguous preset (a=8, b=12, A=35°) and compare both area and angle sets. Expect roughly 47.9 and 19.8 on the two sheets - same SSA list, different finished figures.
06
After the third side exists, Heron’s formula gives area from the semi-perimeter s: √[s(s−a)(s−b)(s−c)]. Perimeter is a+b+c. The measure sheet also tags acute / right / obtuse from the largest angle - a quick shape read, not a full classification course.
The label sketch is schematic (not to scale). It exists so opposite pairs stay readable while you compare twins - not as a CAD export.
Ambiguous twins usually differ in area. That alone is a practical reason to show both solutions: boxing one area while your diagram implies the other is a silent grading trap.
07
Example A · SSS 3-4-5. Preset: classic right triangle. Case strip should read SSS; largest angle ≈ 90°. Area should match (3×4)/2 = 6.
Example B · SAS. Preset: sides a,b with included C=60°. Watch the third side appear from the law of cosines, then confirm angles sum to 180°.
Example C · ASA. Preset: angles A,C with side b between them. Third angle fills first; remaining sides follow sines.
Example D · SSA ambiguous. Preset: two measure sheets. Compare angles B and C across twins - the teaching moment of SSA ≠ unique. Note which twin is acute at B versus obtuse at B.
After presets, blank everything and rebuild one homework figure field-by-field. If the case strip disagrees with the case you thought you had, fix the blanks before trusting area.
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Most “wrong triangle” answers are labeling or case errors, not bad arithmetic. Scan this list before you blame Heron or the law of sines.
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Share links are intentional snapshots. Live typing never rewrites the address bar, so a half-edited homework figure does not leak into a bookmarked URL. When you are ready to send a twin pair or a finished SSS check, use Share / copy on the panel.
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Stop here for planar triangle completion and SSA literacy. Hand off for right-triangle shortcuts, symbolic trig, or ratio setups. Spherical triangles, bearing/azimuth surveying packages, and CAD constraint solvers live outside this page’s job.
11
SSA (two sides and a non-included angle opposite one of them) does not always determine a single triangle. In a window of values there can be zero, one, or two triangles that fit the same givens. This page flags that ambiguous case and shows both solutions when two exist.
SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and SSA (including the ambiguous twin when it appears).
Side a is opposite angle A, side b opposite B, and side c opposite C. Enter lengths in any consistent unit; angles are in degrees.
It names the congruence/solving case detected from your filled fields (SSS, SAS, ASA, AAS, or SSA) so you see which given set you actually entered - not just a pile of numbers.
Once all three sides are known, area = √[s(s−a)(s−b)(s−c)] with s the semi-perimeter. Angles come from the law of cosines or sines depending on the case.
No. Values hydrate from the URL on load; the address bar updates only when you use Share or Copy.
LIVE TRIANGLE SOLVE
SSA ≠ unique ↓Scenario presets
Leave unknowns blank. a opposite A, b opposite B, c opposite C. Any consistent length unit.
CASE STRIP · GIVEN SET
Three sides (SSS) determine a unique triangle up to congruence.
Updates live. SSA ≠ unique when two triangles fit. Congruence cases name the givens - not a CAD proof.
Case SSS · solved
Area 6
Perimeter 12
LABEL SKETCH · NOT TO SCALE
MEASURE SHEET
Angle type: right (by largest angle)
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