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Triangle Solver

Solve any triangle — find missing sides and angles from SSS, SAS, ASA, AAS, or SSA inputs using the law of cosines and sines.

About Triangle Solver

The Triangle Solver computes all missing sides, angles, area, perimeter, and height of any triangle given sufficient information. It supports all five standard input cases: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), AAS (two angles and non-included side), and SSA (two sides and non-included angle — the ambiguous case where there may be 0, 1, or 2 solutions). The solver uses the Law of Cosines for SSS and SAS cases and the Law of Sines for ASA and AAS, and explains which law was applied in the working. For the SSA ambiguous case, all valid solutions are shown. The Law of Cosines reduces to the Pythagorean theorem when C = 90° (cos 90° = 0), confirming it as a generalization. The Law of Sines, combined with the inscribed angle theorem, also gives the circumradius R = a/(2sin A), useful in geometry and coordinate transformations.

Why use Triangle Solver

  • Handles all five standard triangle input cases including ambiguous SSA.
  • Uses the correct law (cosines vs sines) automatically and explains the choice.
  • Shows full solution: sides, angles, area, perimeter, and height.
  • Step-by-step working suitable for geometry and trigonometry coursework.
  • Handles every standard input case rather than requiring you to pick the right formula manually.
  • Privacy-first: side lengths and angles are processed entirely in your browser.

How to use Triangle Solver

  1. Select the input case (SSS, SAS, ASA, AAS, or SSA) from the dropdown.
  2. Enter the known values (sides in any unit, angles in degrees).
  3. Click Solve to see all computed sides, angles, area, and perimeter.
  4. Expand 'Show working' to see the law of cosines/sines derivation.
  5. Read the SSA result carefully — when two solutions exist, decide which is geometrically valid based on the problem context.
  6. Use the area output to cross-check against an independent calculation (e.g., ½ × base × height) when verifying.

When to use Triangle Solver

  • Solving geometry problems in mathematics coursework.
  • Finding missing measurements in surveying or construction applications.
  • Calculating triangle dimensions in engineering or physics problems.
  • Verifying hand-calculated triangle solutions.
  • Surveying or land measurement when only some sides and angles are accessible.
  • Verifying geometry textbook answers and showing complete solution working.

Examples

SSS: a=5, b=6, c=7

Input: Sides: a=5, b=6, c=7

Output: A=44.4°, B=57.1°, C=78.5°; Area≈14.70; Perimeter=18

SAS: a=8, b=10, included angle C=60°

Input: a=8, b=10, C=60°

Output: c≈9.17; A≈49.1°, B≈70.9°; Area≈34.64

SSA ambiguous: a=7, b=10, A=30°

Input: a=7, b=10, A=30°

Output: Two solutions — triangle 1: B=45.6°, C=104.4°, c=13.55; triangle 2: B=134.4°, C=15.6°, c=3.77

Tips

  • Pick the input case that matches what you actually know — entering placeholder values for an SSS solve when you only know SAS will produce a wrong triangle.
  • Always sanity-check that interior angles sum to 180° in the result — small input errors often surface as a 1-2° discrepancy.
  • For SSA (ambiguous case), examine both potential triangles before choosing — context (geometry vs surveying problem) usually rules out one solution.
  • Convert between radians and degrees explicitly — most trig calculators default to one or the other and silently producing nonsense from a unit mismatch is common.
  • If you only need the area and have all three sides, Heron's formula is faster than computing an angle first.
  • Verify the triangle inequality before solving SSS — if any side is ≥ the sum of the other two, no triangle exists.

Frequently Asked Questions

What is the Law of Cosines?▾
c² = a² + b² − 2ab·cos(C). It generalizes the Pythagorean theorem to non-right triangles and is used when three sides or two sides and the included angle are known.
What is the Law of Sines?▾
a/sin(A) = b/sin(B) = c/sin(C). It is used when two angles and a side (or two sides and a non-included angle) are known.
What is the SSA ambiguous case?▾
SSA (two sides and a non-included angle) can produce zero, one, or two valid triangles depending on the relative lengths of the sides. All valid solutions are shown.
How is the triangle area calculated?▾
Area = ½ × a × b × sin(C), where C is the angle between sides a and b. For SSS input, Heron's formula is used: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
Does it solve right triangles?▾
Yes. A right triangle with a 90° angle can be entered as AAS (or SSS if all three sides are known). Pythagoras is implicitly applied and highlighted.
How does this compare to a graphing calculator's triangle solver?▾
Most TI calculators require manual selection of the formula and don't handle the SSA ambiguous case automatically. This tool detects 0, 1, or 2 solutions for SSA and presents all valid triangles with their working.
What if my inputs are in radians?▾
The tool accepts degrees by default; toggle the input mode to radians if your source uses them. Mixing units silently is the most common error in trig solvers.
Can it solve right triangles?▾
Yes — input a 90° angle in any case (AAS or ASA work cleanly with one angle = 90°) and the solver applies the relevant trig including implicit Pythagoras for SSS with a right angle.

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Glossary

Law of Cosines
c² = a² + b² - 2ab·cos(C). Generalizes Pythagoras to non-right triangles. Used for SSS and SAS solving.
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. Used for ASA, AAS, and SSA cases.
Hypotenuse
The side opposite the right angle in a right triangle; always the longest side.
Heron's formula
Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. Computes triangle area from side lengths alone.
Triangle inequality
Each side of a triangle must be less than the sum of the other two: a + b > c, b + c > a, a + c > b.
Ambiguous case (SSA)
When two sides and a non-included angle are given; can yield 0, 1, or 2 valid triangles depending on side ratios.
Altitude / height
Perpendicular distance from a vertex to the opposite side. h_a = 2·Area / a (and similarly for b, c).