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Triangle Calculator

Solve any triangle from three known values (SSS, SAS, ASA, AAS or SSA). Get all sides and angles, the area, the perimeter and the triangle type, with the law of sines or cosines worked out.

  • Free, no sign-up
  • Runs in your browser
  • Math
What do you know?
You know all three sides.
Side a is opposite angle A, side b opposite angle B and side c opposite angle C. Angles are in degrees; use one length unit for all sides.

How to use the calculator

Choose what you know about the triangle, fill in the three boxes that appear and press Solve. The result table lists all three sides, all three angles in degrees, the area, the perimeter and the triangle type. The details panel names the law used and shows every substitution.

The labels follow the usual convention: side a is opposite angle A, side b is opposite angle B and side c is opposite angle C.

  • SSS: three sides.
  • SAS: two sides and the angle between them (a, b and C).
  • ASA: two angles and the side between them (A, B and c).
  • AAS: two angles and a side that is not between them (A, B and a).
  • SSA: two sides and an angle that is not between them (a, b and A).

The methods

Law of cosines connects three sides and one angle: c² = a² + b² − 2ab × cos C. It finds the third side in SAS and every angle in SSS.

Law of sines says the ratio of a side to the sine of its opposite angle is the same for all three sides: a / sin A = b / sin B = c / sin C. It solves ASA, AAS and SSA.

Angle sum: the three angles always add up to 180°, so the last angle is 180° minus the other two.

Worked example (SAS)

Sides a = 8 and b = 11 with angle C = 60° between them:

  1. c² = 8² + 11² − 2 × 8 × 11 × cos 60° = 64 + 121 − 88 = 97, so c ≈ 9.8489
  2. cos A = (11² + 97 − 8²) / (2 × 11 × 9.8489) ≈ 0.71074, so A ≈ 44.70°
  3. B = 180° − 44.70° − 60° ≈ 75.30°
  4. Area = ½ × 8 × 11 × sin 60° ≈ 38.105

All sides differ and every angle is below 90°, so this is a scalene, acute triangle.

The ambiguous SSA case

With two sides and a non-included angle, there may be no triangle, one triangle or two different triangles. For a = 8, b = 10 and A = 40°, sin B = 10 × sin 40° / 8 ≈ 0.8035. Both B ≈ 53.46° and B ≈ 126.54° leave room for a third angle, so two triangles fit: one with c ≈ 12.423 and one with c ≈ 2.898. The calculator shows both side by side. If side a is shorter than b × sin A, no triangle exists.

Good to know

  • The sides must satisfy the triangle inequality: the longest side has to be shorter than the other two together.
  • Angles are in degrees and must be greater than 0° and less than 180°.
  • Results show up to 10 significant digits. A triangle counts as right when its largest angle is 90° to within a tiny rounding margin.

Frequently asked questions

What do SSS, SAS, ASA, AAS and SSA mean?

The letters list what you know, in order around the triangle: S for a side and A for an angle. SAS means two sides with the angle between them; AAS means two angles and a side that is not between them. Each combination fixes the triangle, except SSA, which can have zero, one or two solutions.

Why does SSA sometimes give two triangles?

The given side opposite the known angle can swing to meet the third side at two different points. The law of sines then gives a sine value that matches two angles, one acute and one obtuse. If both leave room for the third angle, both triangles are valid.

Can I solve a triangle from three angles?

No. Three angles fix the shape but not the size: every scaled copy has the same angles. You need at least one side length as well.

How is the triangle type decided?

By sides: equilateral if all three are equal, isosceles if two are equal, scalene otherwise. By angles: right if the largest angle is 90°, obtuse if it is larger and acute if all angles are below 90°.