Solve any triangle by SSS, SAS, ASA, or AAS — find all sides, angles, area, and perimeter.
Three sides
Select what you know — SSS, SAS, ASA, AAS or SSA — and enter the values. Angles are in degrees. The calculator checks that a triangle is possible, then solves for every remaining side and angle, plus area, perimeter, and the in- and circumradius. SSA can have two valid triangles (the "ambiguous case"); both are shown.
Law of Cosines: c² = a² + b² − 2ab·cos(C)
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Heron's formula: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2
Angle sum: A + B + C = 180°. Triangle inequality: each side < sum of the other two.
These are the known-value cases: Side-Side-Side, Side-Angle-Side, Angle-Side-Angle and Angle-Angle-Side. Each determines exactly one triangle when the values are valid.
With two sides and a non-included acute angle, the side opposite the angle can swing to two different positions, producing two valid triangles — or none if it is too short. This is called the ambiguous case.
For SSS: use the Law of Cosines to find each angle. For SAS: use the Law of Cosines to find the third side, then Law of Sines for remaining angles. For ASA/AAS: find the third angle (they sum to 180°), then use Law of Sines to find missing sides. This calculator automatically detects your input type and applies the correct method.
Did you know?
The sum of angles in a triangle always equals exactly 180° in Euclidean geometry. However, on a curved surface (like Earth), triangle angles can sum to more than 180° — a concept central to Einstein's general relativity.
Triangle mathematics dates back to ancient Babylon and Egypt. The Pythagorean theorem (a² + b² = c² for right triangles) was known to Babylonians by 1800 BCE, over a thousand years before Pythagoras. The Law of Cosines generalizing this to all triangles was developed by Persian mathematician al-Kashi in the 15th century.
Heron's formula calculates triangle area from three side lengths without needing the height: Area = √(s(s-a)(s-b)(s-c)) where s is the semi-perimeter.
The strongest structural shape in engineering is the triangle because it cannot be deformed without changing the length of a side.
There are four standard methods to solve any triangle: SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), and AAS (two angles + non-included side).