Solve any triangle using 3 sides or 2 sides + included angle. Computes all angles, area, perimeter, altitudes, medians, circumradius, inradius, and classifies by angles and sides.
A scalene triangle has all three sides of different lengths and all three angles of different measures. It is the most general type of triangle, and every triangle is either scalene, isosceles (two equal sides), or equilateral (all equal). Solving a general triangle — finding every angle, side, area, and derived quantity — requires the Law of Cosines and the Law of Sines.
Given three sides a, b, c (SSS), you can find each angle using the Law of Cosines: cos A = (b² + c² − a²) / (2bc). The area follows from Heron's formula: A = √[s(s−a)(s−b)(s−c)] where s = (a + b + c)/2. Given two sides and the included angle (SAS), the third side comes from c² = a² + b² − 2ab·cos C, and the area is ½ab·sin C.
Once all three sides and angles are known, a host of derived quantities follow: altitudes h_a = 2A/a, medians m_a = ½√(2b² + 2c² − a²), the circumradius R = a/(2 sin A), the inradius r = A/s, and the centroid (intersection of medians). The triangle can be classified as acute, right, or obtuse based on the largest angle, and as scalene, isosceles, or equilateral by its sides.
This calculator supports both SSS and SAS input modes, computes every important property, classifies the triangle, and presents the results with a visual bar chart, a complete properties table, and a reference table of well-known triangles for comparison.
A general triangle solver is useful any time you know enough measurements to define a triangle but still need the rest of the geometry. In surveying, roof framing, truss layout, and site planning, you often start with side lengths or two sides plus an included angle, then need area, remaining angles, or radii without redoing the same trigonometry by hand. This calculator keeps those derived values together so you can move from measurements to decisions faster.
It is also valuable for checking classification and consistency. The outputs make it obvious whether a triangle is acute, right, or obtuse, whether the side lengths satisfy the triangle inequality, and how far the inradius and circumradius sit from the side lengths themselves. That makes it useful both for classroom work and for practical geometry where a wrong angle or misread side can create expensive layout errors.
Law of Cosines: c² = a² + b² − 2ab·cos C Law of Sines: a/sin A = b/sin B = c/sin C = 2R Heron's Formula: A = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2 Area (SAS): A = ½ ab sin C Altitude: h_a = 2A / a Median: m_a = ½√(2b² + 2c² − a²) Circumradius: R = a / (2 sin A) Inradius: r = A / s
Result: Side c ≈ 7.21, area ≈ 20.78 cm², perimeter ≈ 21.21 cm, classification = acute scalene
With SAS input, the calculator first uses the Law of Cosines: c² = 8² + 6² − 2·8·6·cos 60° = 52, so c ≈ 7.21. The area then comes from ½ab sin C = 0.5 × 8 × 6 × sin 60° ≈ 20.78 cm². After the third side is known, the remaining angles, perimeter, medians, altitudes, circumradius, and inradius all follow automatically, giving a complete solution for the triangle.
Unlike equilateral or isosceles cases, a scalene triangle gives you no symmetry shortcuts. Every side and every angle can be different, so the correct method depends on what you know. If all three sides are known, Heron's formula and the Law of Cosines are the natural tools. If two sides and the included angle are known, SAS relationships let you recover the missing side first and then solve the rest. This calculator brings those routes into one workflow instead of forcing you to switch formulas manually.
Solving the triangle is only the beginning. The altitudes show the effective heights relative to each base, the medians locate the balance structure of the triangle, and the inradius and circumradius connect the triangle to its inscribed and circumscribed circles. Those values matter in construction layout, finite-element meshing, and drafting because they describe shape behavior, not just size. A triangle with the same perimeter as another can still have very different area, radii, and stability depending on its angle spread.
Use SSS mode when you have measured all three edges directly, such as from a site sketch or known side lengths in a textbook problem. Use SAS mode when two sides and the included angle are the reliable measurements, which is common in surveying and mechanical linkage work. In both cases, the calculator helps confirm whether the data defines a valid triangle and then expands that minimum input into a full set of geometric properties.
A triangle with all three sides of different lengths (and all three angles different). It is the most general type.
Use the Law of Cosines to find each angle: cos A = (b² + c² − a²) / (2bc). Then use Heron's formula for the area.
c² = a² + b² − 2ab·cos C. It generalizes the Pythagorean theorem — when C = 90° it reduces to c² = a² + b².
Area = ½ × a × b × sin(C), where C is the angle between sides a and b. Use this as a practical reminder before finalizing the result.
A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2. It computes area from the three side lengths alone.
SSS means you know all 3 sides. SAS means you know 2 sides and the included angle. Both uniquely determine the triangle.