Solve any triangle using trigonometric functions. Enter sides and angles to find all missing measurements using Law of Sines, Law of Cosines, and standard trig identities.
The Trigonometric Triangle Solver is a comprehensive tool for finding all unknown sides, angles, and properties of any triangle. Whether you know three sides (SSS), two sides and the included angle (SAS), two angles and the included side (ASA), or two angles and a non-included side (AAS), this calculator applies the Law of Sines and Law of Cosines to determine every remaining measurement. Trigonometric triangle solving is fundamental in surveying, navigation, physics, engineering, and architecture. The Law of Cosines generalizes the Pythagorean theorem to all triangles: c² = a² + b² − 2ab·cos(C). The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C), providing a proportional relationship between sides and opposite angles. Beyond the six basic measurements (three sides and three angles), this calculator also computes the triangle area using the cross-product formula (½·a·b·sin(C)), the circumradius (R = a/(2·sin(A))), and the inradius (r = Area/s where s is the semi-perimeter). Use the preset buttons to explore classic triangle configurations like the 3-4-5 right triangle, equilateral triangles, and isosceles examples. The reference table shows all relevant trig formulas at a glance.
Triangle solving is one of the places where trigonometry becomes genuinely practical: you often know only part of a figure and need the rest. This calculator switches cleanly among SSS, SAS, ASA, and AAS setups, then applies the Law of Sines or Law of Cosines as appropriate. It is useful because it does more than return missing sides and angles; it also reports derived quantities like area, circumradius, and inradius, which helps you verify whether the solved triangle is consistent from multiple perspectives.
Law of Cosines: c² = a² + b² − 2ab·cos(C). Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·a·b·sin(C). Circumradius R = a/(2·sin(A)). Inradius r = Area / s, where s = (a+b+c)/2.
Result: The missing side is about 7.07, the area is about 24.75, and the remaining angles are about 44.4° and 90.6°.
Given a = 7, b = 10, C = 45°: c² = 49 + 100 − 2·7·10·cos(45°) = 149 − 98.99 ≈ 50.01, so c ≈ 7.07. Area = ½·7·10·sin(45°) ≈ 24.75. Using Law of Sines: A = arcsin(7·sin(45°)/7.07) ≈ 44.4°, B ≈ 90.6°.
Not every triangle starts with the same information, so there is no single universal formula for solving one. SSS problems lean on the Law of Cosines first, SAS uses the included angle to unlock a missing side, and ASA or AAS usually begin by finishing the angle sum before applying the Law of Sines. A solver that recognizes the setup saves time and reduces the chance of applying the wrong theorem to the given data.
Once a triangle is solved, there are still more useful measurements available. Area links the side-angle data to physical size, the circumradius describes the unique circle through all three vertices, and the inradius describes the inscribed circle tangent to all three sides. Those extra values are helpful in design, surveying, and geometry proofs because they connect the same triangle to circles, perimeters, and scale.
Working through preset triangles helps you see patterns that are hard to notice from formulas alone. Right triangles show the close relationship between trigonometry and the Pythagorean theorem, while nearly obtuse cases show how sensitive the Law of Cosines can be to angle changes. Comparing several modes with the same output structure builds confidence that different data sets can still describe one consistent triangle.
The Law of Cosines relates the lengths of a triangle's sides to the cosine of one of its angles: c² = a² + b² − 2ab·cos(C). It generalizes the Pythagorean theorem.
The Law of Sines states that the ratio of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). Use this as a practical reminder before finalizing the result.
When given two sides and an angle opposite one of them (SSA), two different triangles may satisfy the conditions. This calculator flags ambiguous cases.
The circumradius R of a triangle is R = a/(2·sin(A)), or equivalently R = (a·b·c)/(4·Area). Keep this note short and outcome-focused for reuse.
The inradius r = Area / s, where s is the semi-perimeter (a+b+c)/2. Apply this check where your workflow is most sensitive.
Yes. Enter two sides or a side and an angle. The calculator detects 90° angles and applies simplified formulas automatically.