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Triangle Solver

Math Calculators

Solve sides and angles (SSS/SAS) with law of cosines

Quick Answer

Triangle Solver: Solve sides and angles (SSS/SAS) with law of cosines. SSS: cos A = (b² + c² − a²) / 2bc; area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2.

Area · Right triangle

6.000

Perimeter 12.000 · SSS

a3.000
b4.000
c5.000
A36.87°
B53.13°
C90.00°

Steps

  1. 1cos A = (b²+c²−a²)/(2bc) → A = 36.87°
  2. 2cos B = (a²+c²−b²)/(2ac) → B = 53.13°
  3. 3C = 180° − A − B = 90.00°
  4. 4Heron: s = 6.00, S = √[s(s−a)(s−b)(s−c)] = 6.000

Which law to use

KnownToolAmbiguity
SSS ←Law of cosines + HeronUnique
SASLaw of cosines (c first)Unique
ASA / AASAngles sum + law of sinesUnique
SSALaw of sines0 / 1 / 2 solutions (ambiguous case)

💡 Three independent pieces (with at least one side) determine a triangle — except SSA, the "ambiguous case": given sides a < b and angle A, you may get zero, one, or two triangles depending on whether a reaches the altitude b·sinA. Surveyors avoid SSA for exactly this reason. The 3-4-5 right triangle is the classic integer Pythagorean triple used to square corners on construction sites.

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How to Use

  1. Enter values or expressions: type directly into the Triangle Solver — keyboard friendly.
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  3. Follow the steps: the result panel shows full working and formula substitution.
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Key Points

  • Triangle Solver is a free online math tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: Solve sides and angles (SSS/SAS) with law of cosines.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Triangle solver (law of cosines)

SSS: cos A = (b² + c² − a²) / 2bc; area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2
SAS: c = √(a² + b² − 2ab·cos C); area = ½ab·sin C
Requires: any two sides sum to more than the third; angle in 0°–180°

FAQ

How do I use the Triangle Solver?
Open the page and fill in the input fields — results update automatically. Solve sides and angles (SSS/SAS) with law of cosines.
Are the results accurate?
Yes. All calculations use standard mathematical formulas and are verified for accuracy.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required — just open and use.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. All calculations run locally in your browser. Nothing is uploaded or stored.

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated:

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

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