To solve a triangle completely, you need three independent values (as long as at least one is a side length). A triangle has six main elements—three sides and three angles—but they’re not all independent because of the constraint that angles must sum to 180°. The minimum information required depends on which values you have: three sides (SSS); two sides and their included angle (SAS); two angles and any side (ASA/AAS); or two sides and a non-included angle (SSA, which may yield multiple solutions). For right triangles, you only need two values (if at least one is a side) because one angle is already known to be 90°. Our calculator analyzes your inputs and selects the appropriate method automatically, calculating all remaining sides, angles, and additional properties like area, perimeter, heights, and the triangle’s type classification. If you provide insufficient or inconsistent information, the calculator will alert you that a unique solution is not possible.

The Law of Sines and Law of Cosines serve different purposes in triangle calculations. The Law of Sines (sin(A)/a = sin(B)/b = sin(C)/c) establishes proportionality between the sides of a triangle and the sines of their opposite angles. It’s ideal for finding unknown sides when you know angles, or unknown angles when you know sides, particularly when the known values aren’t directly related (not an included angle). The Law of Cosines (c² = a² + b² – 2ab·cos(C)) is a generalization of the Pythagorean theorem and works for any triangle. It’s especially useful when you know two sides and their included angle (SAS) or all three sides (SSS). The Law of Sines can yield ambiguous results in some cases (the SSA case), potentially indicating two possible triangles, while the Law of Cosines always provides a unique solution when properly applied. In practice, these laws are often used together: the Law of Cosines to find one unknown element, followed by the Law of Sines to determine the remaining unknowns. Our calculator applies both laws appropriately based on your input method.

When you encounter two possible triangle solutions, you’re dealing with the “ambiguous case,” which occurs specifically when using the Side-Side-Angle (SSA) method under certain conditions. This ambiguity arises when you know two sides and an angle that is not between them. The geometric reason is straightforward: if you imagine fixing one side and the angle, then trying to place the second side of a given length, there may be two possible positions where it intersects the third side. This happens specifically when: 1) The known angle is opposite the shorter of the two known sides; 2) The sine of the known angle is less than the ratio of the shorter side to the longer side; and 3) The triangle inequality can be satisfied in two different configurations. In practical applications, additional constraints or measurements are typically used to determine which solution is correct. For example, in surveying, physical limitations like terrain features might make only one solution feasible. In physics problems, other physical constraints often eliminate one possibility. Our calculator will alert you when an ambiguous case is detected and provide both potential solutions when applicable, allowing you to apply additional context to determine the correct one.

To determine if three sides can form a valid triangle, you need to apply the Triangle Inequality Theorem. This fundamental principle states that the sum of the lengths of any two sides must be greater than the length of the remaining side. For sides a, b, and c, all three of these conditions must be true: a + b > c, a + c > b, and b + c > a. Additionally, all sides must have positive length (a, b, c > 0). If any of these conditions is violated, it’s physically impossible to construct a triangle with the given sides. Intuitively, this makes sense: if two sides are too short compared to the third side, they won’t be able to meet when trying to form the triangle. Our calculator automatically checks these conditions and will alert you if your inputs can’t form a valid triangle. This validation is particularly important in the SSS method, where all three sides are directly specified. Similar validation checks apply to angles: all angles must be greater than 0° and less than 180°, and their sum must equal exactly 180°. These validation rules ensure that the results represent a physically constructible triangle.

The most important properties of a triangle include both its basic elements and derived characteristics. The fundamental elements are its three sides (typically denoted a, b, c) and three angles (A, B, C). From these, we can determine several significant properties: the area, which measures the space enclosed by the triangle; the perimeter, which is the sum of all side lengths; the semi-perimeter, half the perimeter, used in Heron’s formula; the three heights (altitudes), which are perpendicular distances from each vertex to the opposite side; the three medians, lines from each vertex to the midpoint of the opposite side; the incircle radius (inradius), the radius of the circle inscribed within the triangle; and the circumcircle radius (circumradius), the radius of the circle passing through all three vertices. Additionally, triangles are classified by their sides (equilateral, isosceles, or scalene) and by their angles (acute, right, or obtuse). Our calculator provides all these properties automatically, giving you a comprehensive understanding of your triangle’s characteristics. These properties are crucial in various applications from basic geometry to engineering, physics, and architecture, where triangles serve as fundamental building blocks.