Triangle Calculator

About the Triangle Calculator

A triangle calculator solves any triangle — finding all unknown sides, angles, area, and perimeter — from any combination of known values. Every triangle is completely defined by any three of its six elements (three sides and three angles), as long as at least one of those three elements is a side length. Our free triangle calculator handles all four standard cases: SSS (three sides known), SAS (two sides and included angle), ASA (two angles and included side), and AAS (two angles and non-included side). It applies the Law of Sines, Law of Cosines, and basic angle sum property to find all remaining values, then computes area using Heron's formula or the base-times-height method. Results include all angles in degrees, all side lengths, perimeter, area, triangle type classification (acute, right, obtuse; equilateral, isosceles, or scalene), and the circumscribed and inscribed circle radii.

How It Works

Law of Cosines: c^2 = a^2 + b^2 - 2ab x cos(C). Used for SSS and SAS cases. Example SSS: sides 7, 10, 12. To find angle C opposite side c=12: cos(C) = (7^2 + 10^2 - 12^2) / (2x7x10) = (49+100-144)/140 = 5/140 = 0.0357. C = arccos(0.0357) = 87.9 degrees. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Used for ASA and AAS. Example ASA: angle A=50°, side b=8, angle B=60°. Angle C = 180-50-60 = 70°. Side a = b x sin(A)/sin(B) = 8 x sin(50°)/sin(60°) = 8 x 0.766/0.866 = 7.08. Side c = 8 x sin(70°)/sin(60°) = 8 x 0.940/0.866 = 8.68. Area = 0.5 x a x b x sin(C) = 0.5 x 7.08 x 8 x sin(70°) = 26.62 square units.

Tips & Best Practices

• ✓Triangle inequality theorem: any side of a triangle must be less than the sum of the other two sides. If your three sides do not satisfy this, a valid triangle cannot be formed — the calculator will flag this.
• ✓The ambiguous case (SSA): given two sides and a non-included angle, there may be zero, one, or two possible triangles. The calculator identifies and presents all valid solutions.
• ✓Area shortcut: for a right triangle, area = 0.5 x leg1 x leg2 (the two shorter sides). No height calculation needed.
• ✓Sum of angles: the interior angles of any triangle always sum to exactly 180 degrees — this can be used to find a third angle immediately when two are known.
• ✓Heron's formula: for three known sides a, b, c: s = (a+b+c)/2; Area = sqrt(s(s-a)(s-b)(s-c)). Completely avoids needing any angle measurement.
• ✓Special triangles: 30-60-90 triangles have sides in ratio 1:sqrt(3):2. 45-45-90 triangles have sides in ratio 1:1:sqrt(2). Memorising these ratios saves calculation in many geometry problems.
• ✓Right triangle check: if a^2 + b^2 = c^2 exactly, the triangle has a perfect 90-degree angle opposite side c (Pythagorean theorem).
• ✓Circumscribed circle radius (R) = abc / (4 x Area). Inscribed circle radius (r) = Area / s, where s is the semi-perimeter. These are useful for geometry proofs and construction problems.

Who Uses This Calculator

Geometry students solve triangle problems for all four SSS/SAS/ASA/AAS cases. Surveyors use triangle calculations to determine land boundaries and elevations through triangulation. Navigation: ships and aircraft use triangle calculations (celestial navigation, dead reckoning) to determine position. Structural engineers calculate forces in triangular truss members. Architects compute roof pitch angles and rafter lengths from rise-and-run triangles. Trigonometry students work with the Law of Sines and Law of Cosines. Astronomers use triangulation to calculate stellar parallax distances.