Calculators

Triangle Solver

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

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About this tool

The Triangle Solver computes the missing sides and angles of a triangle, plus its area, perimeter and type, from three known values. Enter what you know into the six fields — sides a, b, c and angles A, B, C in degrees — and it solves as you type. Supported combinations are SSS (all three sides), SAS (two sides and the angle between them: a, b, C or a, c, B or b, c, A), and two angles plus a side when the angles are A and B and the side is a or c (ASA/AAS). SSS and SAS use the Law of Cosines; the angle-side cases use the Law of Sines. Area is calculated with Heron's formula from the three sides, and the result includes a small sketch labelled with the three angles and a classification such as Scalene Acute or Isosceles Right. The Law of Cosines reduces to the Pythagorean theorem when C = 90° (cos 90° = 0), confirming it as a generalization.

Why use it

  • Solves SSS, SAS and two-angles-plus-side triangles without choosing a formula yourself.

  • Uses the Law of Cosines or Law of Sines as appropriate behind the scenes.

  • Shows the full solution: sides, angles, area, perimeter and triangle type.

  • Draws a quick sketch labelled with each angle so you can check the shape.

  • Live results with a one-click Clear for working through several problems.

  • Privacy-first: side lengths and angles are processed entirely in your browser.

How to use

  1. Enter any three known values in the side fields (a, b, c) and angle fields (A, B, C, in degrees), including at least one side.
  2. Use a supported combination: three sides; two sides and the angle between them; or angles A and B with side a or c.
  3. Read the results that appear instantly: all three sides and angles, a labelled sketch, area, perimeter and triangle type.
  4. If you only see 'Enter 3 known values', check that your combination is supported and that the values form a real triangle.
  5. Press Clear to empty all six fields before starting a new problem.
  6. Use the area output to cross-check against an independent calculation (e.g., ½ × base × height) when verifying.

When it helps

  • 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

Expected result

A=44.4153°, B=57.1217°, C=78.463°; Area≈14.6969; Perimeter=18; Scalene Acute

SAS: a=8, b=10, included angle C=60°

Input

a=8, b=10, C=60°

Expected result

c≈9.16515; A≈49.1066°, B≈70.8934°; Area≈34.641; Perimeter≈27.1652

AAS: A=40°, B=60°, a=10

Input

A=40°, B=60°, a=10

Expected result

C=80°; b≈13.473, c≈15.3209; Area≈66.3414; Perimeter≈38.7939

Tips

  • Pick values that match what you actually know — entering placeholder numbers in extra fields can produce a different triangle.
  • Always sanity-check that interior angles sum to 180° in the result — small input errors often surface as a 1-2° discrepancy.
  • For an SSA problem (two sides and a non-included angle), first find a third value by hand with the Law of Sines, then enter a supported combination.
  • Convert radians to degrees before entering angles — the angle fields only accept degrees.
  • 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). The solver uses it when two angles and a side are known.
Which combinations of values can I enter?
Three sides (SSS); two sides with the angle between them (a, b, C or a, c, B or b, c, A); or angles A and B with side a or c. Other combinations, such as two sides and a non-included angle (SSA), are not solved.
How is the triangle area calculated?
Once all three sides are known, the tool uses Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
Does it solve right triangles?
Yes. Enter all three sides, or two legs with a 90° included angle, and the type will read Right (for example Scalene Right for a 3-4-5 triangle).
Why does it say 'Enter 3 known values' after I typed three?
Either the combination is not one the solver supports, or the numbers cannot form a triangle — for example two angles adding up to 180° or more, or one side longer than the other two combined.
What if my inputs are in radians?
Angles must be entered in degrees. Convert radians first by multiplying by 180/π (about 57.2958).
How is the triangle type decided?
By sides it is Equilateral, Isosceles or Scalene; by angles it is Right (largest angle within 0.1° of 90°), Obtuse or Acute.

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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).
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