Triangle Calculator

The triangle inequality states that the sum of any two sides must be greater than the third side. If a + b ≤ c, no triangle can exist because the two shorter sides cannot reach across the longest side. This is why SSS input accepts any three positive numbers but may return "no valid triangle" if they violate this fundamental geometric constraint. For example, sides 1, 2, and 5 cannot form a triangle because 1 + 2 < 5.

The "included angle" is the angle between the two known sides. Without it, two different triangles could share the same two side lengths — imagine a hinge opening to different angles. SAS uniquely determines a triangle because fixing the angle between the two known sides forces a specific third side length via the law of cosines. If you have two sides and a non-included angle, that is the SSA case which can produce zero, one, or two possible triangles.

Use SSS if you have all three side lengths (common when measuring a physical triangle). Use SAS if you have two sides and the angle between them (common in construction and surveying). Use ASA if you have two angles and the side between them (common in navigation and triangulation). These three patterns cover the most common real-world measurement scenarios. The AAS case (two angles, non-included side) is equivalent to ASA since the third angle can always be found.

The SVG renders the triangle to scale based on the computed side lengths. It may appear rotated or flipped differently than your mental image — this is because the canonical coordinate system places vertex C at the origin and vertex B along the positive x-axis for consistent rendering. The shape, proportions, and all angles are geometrically accurate regardless of orientation.

Heron's formula computes the area of a triangle using only its three side lengths: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. This is useful when you know all three sides but not the height. It was discovered by Heron of Alexandria and works for any triangle. For SSS mode, this is the primary method for calculating area. For SAS and ASA modes, area can also be computed directly using A = ½ab·sin(C) since you know two sides and the included angle.