Triangle Side and Angle Solver: Fast Steps to Complete Triangle Solutions

Ultimate Triangle Side and Angle Solver: From SSA to SSS Explained—

Understanding triangles is central to geometry, trigonometry, physics, engineering and many everyday problems. Whether you’re solving classroom exercises, working on surveying problems, or programming a geometry tool, knowing how to determine missing sides and angles of a triangle from various known elements is essential. This article explains the main triangle-solving cases—SSS, SAS, ASA, AAS, and SSA—provides step-by-step solution methods, highlights pitfalls (especially the ambiguous SSA case), and gives worked examples and formulas you can apply.

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Overview: The Five Classical Cases

Triangles are typically solved when three independent pieces of information are known. Those pieces can be sides (S) or angles (A). The common cases are:

• SSS (Side–Side–Side): three sides known — unique triangle.
• SAS (Side–Angle–Side): two sides and included angle known — unique triangle.
• ASA (Angle–Side–Angle): two angles and the included side known — unique triangle.
• AAS (Angle–Angle–Side): two angles and a non-included side known — unique triangle.
• SSA (Side–Side–Angle or the “ambiguous case”): two sides and a non-included angle known — may produce zero, one, or two triangles.

Key fact: given three independent elements (with at least one side), a triangle can be determined up to congruence. The SSA case is the only one that can be ambiguous.

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Tools and Formulas You’ll Use

• Law of Cosines: For sides a, b, c opposite angles A, B, C: c^2 = a^2 + b^2 − 2ab cos C (and cyclic permutations)

• Law of Sines: sin A / a = sin B / b = sin C / c

• Angle sum: A + B + C = 180°

• Area formulas:

  • (⁄₂)ab sin C (useful when two sides and included angle are known)
  • Heron’s formula for area from sides a, b, c: s = (a + b + c)/2 Area = sqrt[s(s − a)(s − b)(s − c)]

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Case-by-Case Methods and Examples

1) SSS (Side–Side–Side)

When all three sides a, b, c are known, the triangle is uniquely determined.

Method:

• Use the Law of Cosines to find one angle (e.g., C): cos C = (a^2 + b^2 − c^2) / (2ab) then C = arccos(…)
• Find other angles similarly or use angle sum.

Example: a = 7, b = 8, c = 9 cos C = (7^2 + 8^2 − 9^2)/(2·7·8) = (49 + 64 − 81)/112 = ⁄₁₁₂ = ⁄₇
C = arccos(⁄₇) ≈ 73.4° Then find A, B by permutation or angle sum.

2) SAS (Side–Angle–Side)

Two sides and the included angle determine a unique triangle.

Method:

• Use Law of Cosines to find the third side, or use Law of Sines to find other angles after computing the third side.

Example: a = 5, b = 6, C = 60° c^2 = a^2 + b^2 − 2ab cos C = 25 + 36 − 60·0.5 = 61 − 30 = 31 c = sqrt(31) ≈ 5.567 Then use Law of Sines to find other angles.

3) ASA (Angle–Side–Angle)

Two angles and the included side determine a unique triangle.

Method:

• Compute the third angle by angle sum.
• Use Law of Sines to compute the remaining sides.

Example: A = 50°, C = 60°, b = 10 (side between angles A and C) B = 70° Use Law of Sines: a/sin A = b/sin B → a = b·sin A / sin B, etc.

4) AAS (Angle–Angle–Side)

Two angles and any side produce a unique triangle (equivalent to ASA in practice).

Method:

• Same as ASA: find third angle, then use Law of Sines.

Example: A = 40°, B = 55°, c = 12 C = 85° Use Law of Sines to find a and b.

5) SSA (Side–Side–Angle) — The Ambiguous Case

Given two sides (a, b) and an angle A that is not between them (for example, angle A opposite side a), the number of possible triangles depends on the values.

Let given: side a, side b, and angle A (opposite a). Let h = b·sin A (the altitude from angle B if you place side b as base).

• If a < h: no triangle (side a is too short to reach base).
• If a = h: one right triangle (a is exactly the altitude).
• If h < a < b: two distinct triangles (one acute, one obtuse) — ambiguous case.
• If a ≥ b: one triangle (a long enough to span).

Method:

• Use Law of Sines to find angle B: sin B = b·sin A / a If RHS > 1 → no solution. If RHS = 1 → B = 90° (unique). If 0 < RHS < 1 → two possible B: B1 = arcsin(RHS), B2 = 180° − B1 (if B2 is consistent with angle sum).
• For each valid B, compute C = 180° − A − B, then find remaining side(s) by Law of Sines.

Example: Given A = 30°, a = 4, b = 7: h = b·sin A = 7·0.5 = 3.5. Since a = 4 > h and a < b, expect two solutions. Compute sin B = b·sin A / a = 7·0.5 / 4 = 0.875 B1 = arcsin(0.875) ≈ 61.0° B2 = 180° − 61.0° = 119.0° Then C1 = 180 − 30 − 61 = 89°, C2 = 31° Use Law of Sines to get corresponding c values.

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Practical Tips and Common Pitfalls

• Always check which angle a given side is opposite to; mixing up labels causes wrong application of the laws.
• When using inverse sine in SSA, consider the possibility of two solutions: arcsin returns the acute angle; check 180° − arcsin(…) as a second candidate.
• Round-off error can misclassify RHS values slightly above 1; clamp values like 1.00000001 down to 1 when it’s within numerical tolerance.
• Use Law of Cosines for robust determination when angles are near 0° or 180° (sin becomes small and less stable).
• For programming solvers, systematically handle edge cases: degenerate triangles (sum of two sides equal the third), zero/negative lengths, and floating-point tolerances.

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Worked Examples (Step-by-step)

Example 1 — SSS: a = 6, b = 7, c = 8 cos C = (6^2 + 7^2 − 8^2)/(2·6·7) = (36 + 49 − 64)/84 = ⁄₈₄ = 0.25 C = arccos(0.25) ≈ 75.52° A from cos A = (b^2 + c^2 − a^2)/(2bc) etc., then B = 180 − A − C.

Example 2 — SSA ambiguous: A = 40°, a = 5, b = 9 h = b·sin A = 9·sin 40° ≈ 9·0.643 = 5.787 a = 5 < h → no triangle.

Example 3 — SAS: a = 10, b = 7, C = 120° c^2 = 10^2 + 7^2 − 2·10·7·cos 120° = 100 + 49 − 140·(−0.5) = 149 + 70 = 219 c ≈ 14.817

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Algorithms for Implementing a Triangle Solver

1. Validate inputs: positive side lengths, angles between 0° and 180°, at least one side provided.
2. Normalize: convert degrees ↔ radians as needed; standardize labels so angle given is opposite side a when using SSA logic.
3. Branch on case type (count of known sides/angles and whether known angle is included).
4. Apply appropriate formulas (Law of Cosines or Law of Sines).
5. For SSA, compute h and handle 0/1/2 solutions; verify solutions by recomputing consistency (sides, angles sum).
6. Return results with tolerances and flags for ambiguous/no solution/degenerate triangle.

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Summary and Quick Reference

• SSS, SAS, ASA, AAS → unique triangle.
• SSA → 0, 1, or 2 triangles depending on the relation between a, b, and b·sin A.
• Use Law of Cosines for sides-first computations; use Law of Sines for angle propagation when safe.
• Check numerical tolerances and ambiguous-case alternatives when using inverse sine.

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