Triangle Solver

Solve any triangle — not just right triangles — from SSS, SAS or ASA: all sides, all angles, the perimeter, the area and the triangle type.

Side a is opposite angle A

Result

Enter values and press Calculate to see the result.

Formula

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

The law of cosines links three sides and one angle.

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

The law of sines: each side over the sine of its opposite angle is the same.

Area=s(s−a)(s−b)(s−c),s=a+b+c2\text{Area} = \sqrt{s(s-a)(s-b)(s-c)},\quad s = \frac{a+b+c}{2}

Heron’s formula gives the area from the three sides.

How to use this calculator

  1. Choose what you know: three sides, two sides and the angle between them, or two angles and the side between them.
  2. Side a is opposite angle A, side b opposite angle B and side c opposite angle C.
  3. Press Calculate to get every side, every angle, the perimeter, the area and the triangle type.

Worked example

A triangle with sides 3, 4 and 5

Check the triangle inequality: 3 + 4 > 5, so the triangle exists.

Law of cosines: cos A = (16 + 25 − 9)/40 = 4/5, so A ≈ 36.869898°; cos B = 3/5, so B ≈ 53.130102°; cos C = 0, so C = 90°.

Heron: s = 6 and the area is √(6 × 3 × 2 × 1) = 6.

Frequently asked questions

Why does it say no triangle exists?

Any two sides must add up to more than the third side, and the angles must add up to exactly 180°.

Why is SSA (two sides and an angle not between them) not offered?

SSA can give two different triangles or none at all — the “ambiguous case”. Sumzo AI can walk through it.