About this calculator
A triangle has six measurements: three sides (a, b, c) and three angles (A, B, C), where each angle sits opposite the side with the same letter. Know the right three and you can find the other three.
Fill in any three boxes, with at least one of them a side, and leave the rest blank. The calculator works out which method applies (SSS, SAS, ASA, AAS or SSA), solves it and shows each step. Angles are in degrees.
Worked examples
Real numbers, worked out by the same calculator. Press “Use these numbers” to try one above.
A 3-4-5 triangle (three sides)
- Angle A
- 36.869898°
- Side a
- 3
- Side b
- 4
- Side c
- 5
- Angle B
- 53.130102°
- Angle C
- 90°
- Area
- 6
- Perimeter
- 12
- Type
- scalene, right-angled
The triangle has sides 3, 4 and 5, angles 36.869898°, 53.130102° and 90°, and an area of 6. It is scalene, right-angled.
Show the working
- Three sides (SSS): use the law of cosines, cos A = (b² + c² − a²) ÷ 2bc, for each angle.
- Area = ½ a b sin C = ½ × 3 × 4 × sin(90°) = 6.
Two sides and the angle between them
- Side c
- 5
- Side a
- 3
- Side b
- 4
- Angle A
- 36.869898°
- Angle B
- 53.130102°
- Angle C
- 90°
- Area
- 6
- Perimeter
- 12
- Type
- scalene, right-angled
The triangle has sides 3, 4 and 5, angles 36.869898°, 53.130102° and 90°, and an area of 6. It is scalene, right-angled.
Show the working
- Two sides and the angle between them (SAS): law of cosines, c² = a² + b² − 2 ab cos C, so c = 5.
- The other angles come from the law of cosines again: A = 36.869898° and B = 53.130102°.
- Area = ½ a b sin C = ½ × 3 × 4 × sin(90°) = 6.
Two angles and a side (30°, 60°, c = 10)
- Side a
- 5
- Side b
- 8.660254
- Side c
- 10
- Angle A
- 30°
- Angle B
- 60°
- Angle C
- 90°
- Area
- 21.650635
- Perimeter
- 23.660254
- Type
- scalene, right-angled
The triangle has sides 5, 8.660254 and 10, angles 30°, 60° and 90°, and an area of 21.650635. It is scalene, right-angled.
Show the working
- Two angles and a side (ASA or AAS): the third angle is 180° − 30° − 60° = 90°.
- The law of sines, a ÷ sin A = b ÷ sin B = c ÷ sin C, gives the sides: each ÷ its sine = 10.
- Area = ½ a b sin C = ½ × 5 × 8.660254 × sin(90°) = 21.650635.
Two sides and a non-included angle (SSA)
- Side c
- 15.556704
- Side a
- 10
- Side b
- 12
- Angle A
- 40°
- Angle B
- 50.474835°
- Angle C
- 89.525165°
- Area
- 59.99794
- Perimeter
- 37.556704
- Type
- scalene, acute
The triangle has sides 10, 12 and 15.556704, angles 40°, 50.474835° and 89.525165°, and an area of 59.99794. It is scalene, acute. A second triangle also fits (see the working).
Show the working
- Two sides and an angle not between them (SSA): law of sines, sin B = b × sin A ÷ a = 12 × sin(40°) ÷ 10, so B = 50.474835°.
- The third angle is 180° − 40° − 50.474835° = 89.525165°, and side c = a × sin C ÷ sin A = 15.556704.
- Area = ½ a b sin C = ½ × 10 × 12 × sin(89.525165°) = 59.99794.
- A second triangle also fits: B = 129.525165°, C = 10.474835° and c = 2.828363. (This is the "ambiguous case" of SSA.)
Which method is used
- SSS (three sides): law of cosines gives each angle
- SAS (two sides and the angle between them): law of cosines gives the third side, then the angles
- ASA or AAS (two angles and a side): the angles add to 180°, then the law of sines gives the sides
- SSA (two sides and an angle not between them): the law of sines, which can give two answers
The formulas
- Law of cosines: c² = a² + b² − 2ab cos C
- Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C
- Angle sum: A + B + C = 180°
- Area: ½ × a × b × sin C
For a right-angled triangle these reduce to Pythagoras and the usual sine, cosine and tangent.
The ambiguous case (SSA)
Give two sides and an angle that is not between them and there can be no triangle, one triangle or two different triangles. Picture swinging the unknown side like a gate: it can touch the base in two places. When two fit, the calculator shows the first and describes the second in the working.
Frequently asked questions
How many values do I need to solve a triangle?
Three, and at least one must be a side. Three angles alone only give the shape, not the size.
What do the letters mean?
Side a is opposite angle A, side b opposite angle B and side c opposite angle C.
Why did it say no triangle fits?
The numbers break a rule: the two shorter sides must add to more than the longest, the angles must add to less than 180°, or the side opposite the angle is too short to reach.
Can it solve right-angled triangles?
Yes. Enter a 90° angle with two other values.
What is the ambiguous case?
With two sides and a non-included angle two different triangles can sometimes fit. The working describes the second one.
Formulas tested against hand-worked answers. Last reviewed 29 September 2026. These calculators do arithmetic only; they are not financial, tax or legal advice.