Trigonometric Triangle Solver

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.

Side a
3.00
Length of side a
Side b
4.00
Length of side b
Side c
5.00
Length of side c
Angle A
36.87°
Angle opposite side a
Angle B
53.13°
Angle opposite side b
Angle C
90.00°
Angle opposite side c
Area
6.00
½ · a · b · sin(C)
Perimeter
12.00
a + b + c
Circumradius (R)
2.50
abc / (4·Area)
Inradius (r)
1.00
Area / semi-perimeter

Side Comparison

a
3.00
b
4.00
c
5.00

Angle Comparison

A
36.87°
B
53.13°
C
90.00°
Trigonometry Formulas Reference
FormulaExpression
Law of Cosinesc² = a² + b² − 2ab·cos(C)
Law of Sinesa/sin(A) = b/sin(B) = c/sin(C)
Area (SAS)½ · a · b · sin(C)
Area (Heron)√(s(s−a)(s−b)(s−c))
CircumradiusR = abc / (4·Area)
Inradiusr = Area / s
Semi-perimeters = (a + b + c) / 2
Angle SumA + B + C = 180°
Planning notes, formulas, and examples

About the Trigonometric Triangle Solver

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.

When This Page Helps

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.

How to Use the Inputs

  1. Select a solve mode: SSS, SAS, ASA, or AAS.
  2. Enter the known sides and/or angles in the input fields.
  3. Click a preset button to load a common triangle configuration.
  4. Review the computed sides, angles, area, circumradius, and inradius.
  5. Consult the formulas reference table for the underlying trig identities.
Formula used
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.

Example Calculation

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°.

Tips & Best Practices

  • For SSS mode, all three sides must satisfy the triangle inequality.
  • Angles are entered in degrees — the calculator converts internally.
  • SAS requires the angle between the two given sides.
  • Check the ambiguous case warning when using AAS — two solutions may exist.

Choosing the Right Solving Path

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.

Beyond Missing Sides and Angles

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.

Building Trig Intuition

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.

Sources & Methodology

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Frequently Asked Questions

  • 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.

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