Triangle Calculator

Enter any two sides (a, b, c) or a side and an angle to solve the triangle. Angles are in degrees.

Triangle Formulas

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Solving Triangles: The Key Formulas

A triangle has three sides (a, b, c) and three angles (A, B, C) that always sum to 180°. Given three pieces of information (at least one side), you can solve the triangle. For a right triangle, use the Pythagorean theorem (a² + b² = c²) and basic trigonometry: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent. For non-right triangles, use the Law of Sines: a/sin A = b/sin B = c/sin C, or the Law of Cosines: c² = a² + b² − 2ab cos C.

Area of a triangle: ½ × base × height (requires a height). Alternative: Heron's formula uses only the three sides — s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)). Triangles appear in surveying, architecture, navigation (triangulation), engineering trusses, and computer graphics (all 3D surfaces are meshes of triangles). The longest side is always opposite the largest angle. An equilateral triangle (all sides equal) has all angles = 60°.

Triangle Classification

Triangle Properties and Key Theorems

Triangles are the simplest polygon and possess several fundamental properties. The interior angles of any triangle always sum to exactly 180 degrees. A right triangle contains one 90-degree angle; the side opposite the right angle is the hypotenuse, and the Pythagorean theorem relates its length to the legs: a squared plus b squared equals c squared. An equilateral triangle has all three sides equal and all three angles equal at 60 degrees. An isosceles triangle has two equal sides and two equal base angles. A scalene triangle has no equal sides or angles. The area of a triangle equals one-half the base times the perpendicular height, regardless of which side you choose as the base. When only side lengths are known, Heron's formula computes area without requiring the height: calculate the semi-perimeter s, then area equals the square root of s times (s minus a) times (s minus b) times (s minus c). The law of sines and the law of cosines extend trigonometry to any triangle, not just right triangles, allowing calculation of unknown sides and angles when partial information is given. Triangles appear in structural engineering as the most rigid polygon, in navigation using triangulation, and in computer graphics as the basic primitive for rendering 3D surfaces.

Triangle Types Reference Table