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Geometric Analysis Engine

Triangle Calculator

Solve any triangle with ease. Enter any three values (sides or angles) to instantly calculate all remaining parameters with geometric precision.

C (γ)A (α)B (β)bac

Input RequirementPlease provide 3 positive values, including at least one side length.

Analytical Ready

Input your triangle parameters to see a comprehensive geometrical breakdown.

Geometric Solvers

Decoding
Triangle Geometry

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. Solving a triangle means finding the lengths of all three sides and the measurements of all three angles.

Law of Sines

Relates the sides of a triangle to the sines of its angles. Essential for AAS or SSA scenarios.

a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines

Used to find a side when two sides and the included angle are known (SAS) or all three sides (SSS).

c² = a² + b² - 2ab cos(C)

The Trigonometry of Triangles

To solve any triangle, you generally need to know at least three of its six parameters (3 sides and 3 angles), and at least one of those must be a side length. Knowing only three angles identifies a set of similar triangles but doesn't fix a specific size.

Triangle Classification

Triangles are classified based on their sides and angles:

  • Equilateral: All three sides are equal and all angles are 60°.
  • Isosceles: Two sides are equal and two angles are equal.
  • Scalene: All sides and angles are different.
  • Right-angled: One angle is exactly 90°.
  • Obtuse: One angle is greater than 90°.
  • Acute: All angles are less than 90°.

Heron's Formula for Area

When only the lengths of all three sides are known, Heron's formula is used to calculate the area without needing the height:

s = (a + b + c) / 2
Area = √[ s(s-a)(s-b)(s-c) ]

Standard Input Cases:

  • • SSS: Known three sides. Always solvable if the sum of two sides > third side.
  • • SAS: Two sides and the angle between them. Always solvable.
  • • ASA / AAS: Two angles and one side. Always solvable.
  • • SSA: The "Ambiguous Case". May result in zero, one, or two valid triangles.

Solving a Right Triangle

In a right triangle, the relationships simplify significantly using SOH CAH TOA:

sin(θ) = Opp / Hyp
cos(θ) = Adj / Hyp
tan(θ) = Opp / Adj

How to Use This Solver

Input any three values (ensuring at least one is a side) into the configuration panel. The engine will:

  1. Detect the input case (SSS, SAS, etc.).
  2. Calculate all missing angles and side lengths.
  3. Determine the Area and Perimeter.
  4. Classify the triangle by sides and angles.
  5. Verify the validity of the inputs using the Triangle Inequality Theorem.