△ Triangle Calculator

Solve any triangle — find sides, angles, area, and perimeter using the Law of Sines and Law of Cosines.

Right Triangle — Pythagorean Theorem

SSS — All Three Sides Known

SAS — Two Sides and Included Angle

Area — Multiple Methods

Key Formulas

  • Pythagorean: c² = a² + b²
  • Law of Cosines: c² = a² + b² − 2ab·cos(C)
  • Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
  • Heron's formula: A = √[s(s−a)(s−b)(s−c)], s=(a+b+c)/2

How the Triangle Calculator Works

How Triangle Solving Works

Triangle solving finds unknown sides and angles using the relationships between known values. Different methods are used depending on which sides and angles are given.

Mathematical Laws & Formulas:

Law of Cosines:
c² = a² + b² – 2ab cos(C)
Use when you know 2 sides and included angle (SAS) or 3 sides (SSS)
Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
Use when you know 2 angles and 1 side (ASA, AAS) or 2 sides and 1 angle (SSA)
Heron’s Formula (Area):
A = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2
Calculate area when all three sides are known

Solution Methods Guide:

  • SSS: All sides → Use Law of Cosines to find angles
  • SAS: 2 sides + included angle → Use Law of Cosines for third side
  • ASA/AAS: 2 angles + 1 side → Use Law of Sines
  • SSA: 2 sides + non-included angle → May have 0, 1, or 2 solutions

Real-World Applications:

  • Surveying: Land measurement, property boundaries
  • Navigation: GPS systems, marine/aviation navigation
  • Construction: Roof trusses, structural engineering
  • Astronomy: Measuring distances to stars using parallax
  • Physics: Force vector analysis, wave interference

Important Notes:

  • Sum of angles in any triangle always equals 180°
  • Triangle inequality: sum of any two sides must be greater than the third
  • SSA case can be ambiguous – may result in multiple valid solutions
  • All angles must be positive and less than 180°
  • Use degrees or radians consistently throughout calculations
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