TriSolve

Triangle Calculator

Three given parts — sides, angles, or a mix — solve the whole triangle, including area.

Side a

Side b

Side c

Side a: 3
Side b: 4
Side c: 5
Angle A: 36.87°
Angle B: 53.13°
Angle C: 90°

Perimeter: 12
Area: 6

Triangles solved from three sides

With all three sides known, the law of cosines gives every angle and Heron's formula gives the area.

a	b	c	Angle A	Angle B	Angle C	Area	Perimeter
3	4	5	36.870	53.130	90.000	6.0000	12.00
5	5	5	60.000	60.000	60.000	10.8253	15.00
6	8	10	36.870	53.130	90.000	24.0000	24.00
7	8	9	48.190	58.412	73.398	26.8328	24.00
10	10	12	53.130	53.130	73.740	48.0000	32.00
5	12	13	22.620	67.380	90.000	30.0000	30.00

Rows one, three and six all produce a 90 degree angle because they are Pythagorean triples - the law of cosines reduces to Pythagoras exactly when the triangle is right-angled. Row two is equilateral, so every angle is 60. Rows two and five are isosceles, and their equal sides produce equal opposite angles. Angles always sum to 180, which is the quickest check on any answer. Three sides only form a triangle if each is shorter than the other two combined; violate that and no triangle exists.

Which laws apply

Law of Cosines handles SSS and SAS cases. Law of Sines handles ASA. Right triangles have shortcuts via Pythagoras and basic trig.

Triangle area formulas

Area = ½ × base × height (if you know the height). Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. SAS formula: Area = ½ × a × b × sin(C). This calculator picks the best formula based on your inputs.

Frequently asked questions

How do I solve a triangle with 3 sides (SSS)?

Use the Law of Cosines to find each angle: cos(A) = (b² + c² − a²) / (2bc). Find the largest angle first (opposite the longest side), then use Law of Cosines or Sines for the second, and subtract both from 180° for the third. Area via Heron's formula: √(s(s−a)(s−b)(s−c)).

What is the Law of Sines?

a/sin(A) = b/sin(B) = c/sin(C). It relates each side to the sine of its opposite angle. Use it when you know an angle-side pair plus one more angle or side. Watch for the ambiguous case (SSA): two different triangles may satisfy the same inputs.

How do I find the area of a triangle?

If you know base and height: Area = ½ × base × height. If you know two sides and the included angle: Area = ½ × a × b × sin(C). If you know all three sides: Heron's formula — semi-perimeter s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)).