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Triangle Solver

Solve any triangle from SSS, SAS, ASA, AAS, SSA, or right-triangle (Pythagoras) inputs. Drawn-to-scale visualization with all sides, angles, area, and perimeter.

Math

Triangle Solver

Generated on April 25, 2026

Know all three sides

Area
6.000
sq units
Perimeter
12.000
units
Type
Right scalene
Largest angle
90.00°

Triangle (drawn to scale)

ABCa = 3.00b = 4.00c = 5.0037°53°90°

All sides & angles

Side a3.0000
Side b4.0000
Side c5.0000
Angle A36.87°
Angle B53.13°
Angle C90.00°
Sum of angles180.00°

Step-by-step calculation

Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C).

  1. 1Mode: SSS.
  2. 2Given values produce a triangle with sides 3.00, 4.00, 5.00.
  3. 3Angles: A=36.87°, B=53.13°, C=90.00° (sum = 180.00°, must be 180°).
  4. 4Area via SAS formula: ½·3.00·4.00·sin(90.00°) = 6.0000 sq units.
  5. 5Perimeter: 3.00 + 4.00 + 5.00 = 12.0000 units.

?What is the Triangle Solver?

The Triangle Solver finds every side, angle, area, perimeter, and triangle classification (acute / right / obtuse, scalene / isosceles / equilateral) from any valid combination of three known values: SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), SSA (two sides + non-included angle — the 'ambiguous case'), or a right triangle (Pythagoras). The result is rendered as a drawn-to-scale 2D triangle with side and angle labels, so you can verify the geometry visually. Uses the Law of Sines, Law of Cosines, and Heron's formula under the hood.

The Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C). Pythagoras (right triangle): c² = a² + b².

The Law of Cosines generalizes Pythagoras: when C = 90°, cos(C) = 0 and the formula reduces to c² = a² + b². For any other angle, the −2ab·cos(C) term adjusts for the angle. The Law of Sines is used when you know an angle and its opposite side. The SSA case is 'ambiguous' because the geometry can sometimes form two valid triangles — when this happens, the calculator reports both. Heron's formula gives area from three sides alone: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.

Practical Examples

1

3-4-5 right triangle: SSS check — area 6, angles 36.87° / 53.13° / 90°.

2

Surveying: from two known angles and a baseline distance (ASA), compute the distance to a far point.

3

Carpentry: cutting a roof rafter — given run + rise, find the hypotenuse and angle.

4

Navigation: given two bearings to a landmark and the distance between observation points (AAS), triangulate the landmark's position.

5

Equilateral triangle (60-60-60): every triangle with all three angles equal is also equilateral (all sides equal).

6

Sum of any two sides MUST exceed the third — the triangle inequality. The calculator catches violations.

Frequently Asked Questions

When you know two sides and an angle that's NOT between them, sometimes two different triangles satisfy the constraints. The calculator detects this and reports both solutions. SSA is the only case in classical trigonometry that can have multiple answers.

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