Triangle Calculator

Given any combination of sides and angles, this triangle calculator solves for all missing values — including area, perimeter, and all three angles — using the law of sines and law of cosines.

How to Use This Calculator

1. Enter the known values: sides (a, b, c) and/or angles (A, B, C).
2. You need at least three values to solve a triangle (including at least one side).
3. Click Calculate to find all remaining sides, angles, area, and perimeter.

Key Triangle Formulas

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines: c² = a² + b² − 2ab×cos(C)

Area (from sides & angle): Area = ½ × a × b × sin(C)

Area (Heron's Formula): Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2

Angle sum: A + B + C = 180°

Example: SSA Case

a = 10, b = 15, A = 35° — solve for everything:

• sin(B) = (b × sin(A))/a = (15 × sin35°)/10 = 0.8604 → B ≈ 59.5°
• C = 180° − 35° − 59.5° = 85.5°
• c = a × sin(C)/sin(A) = 10 × sin(85.5°)/sin(35°) ≈ 17.35
• Area: ½ × 10 × 15 × sin(85.5°) ≈ 74.9

Triangle Types

• Equilateral: All sides equal, all angles 60°
• Isosceles: Two sides equal, two angles equal
• Scalene: No sides equal, no angles equal
• Right: One 90° angle (use Right Triangle Calculator)
• Obtuse: One angle > 90°

Common Mistakes to Avoid

• SSA ambiguous case — Given two sides and an opposite angle, there may be 0, 1, or 2 valid triangles. The calculator handles this, but be aware ambiguity exists.
• Using law of cosines with sines and vice versa — Use Law of Sines when you have angle-side pairs. Use Law of Cosines for SSS or SAS configurations.
• Mixing degrees and radians — Set your calculator mode. Law of sines/cosines in degrees requires degree-mode trig functions.

Frequently Asked Questions

What information do I need to solve a triangle?

Any of: SSS (all three sides), SAS (two sides + included angle), AAS (two angles + any side), ASA (two angles + included side), or SSA (two sides + non-included angle — check for ambiguity).

How do I find the area without the height?

Use Heron's formula (three sides known) or the formula ½ab×sin(C) if you know two sides and the included angle.

What is the triangle inequality theorem?

Any two sides must sum to more than the third side. If a + b < c (or any permutation), the triangle doesn't exist. This calculator will flag invalid inputs.

Conclusion

The triangle is the most fundamental polygon in geometry and engineering. Use this calculator to solve any triangle from the minimum required inputs — whether for homework, design, or real-world measurement.

Related: Right Triangle Calculator | Pythagorean Theorem Calculator | Area Calculator | Scientific Calculator