📐 Triangle Solver
Solve triangles using SSS, SAS, or ASA methods - calculate missing sides and angles
Solution Methods
How to Use This Calculator
- Choose the solving method from the dropdown based on what information you have (SSS, SAS, or ASA)
- Enter the known side lengths and angles into the corresponding input fields
- For SSS: Enter all three sides (a, b, c)
- For SAS: Enter two sides and the angle between them
- For ASA: Enter two angles and the side between them
- Click "Solve Triangle" to calculate all missing sides, angles, and area
- Review the complete solution showing all sides (in same units as input) and angles (in degrees)
- Try different methods with the same triangle to verify your results
Understanding Triangle Solving
Triangle Properties and Basics
A triangle is a polygon with three sides and three angles. Several fundamental properties always hold: (1) The sum of all three angles equals exactly 180°. (2) The triangle inequality theorem: the sum of any two sides must be greater than the third side. (3) Each angle is opposite a side—larger angles are opposite longer sides. (4) Triangles are classified by sides (scalene: all different, isosceles: two equal, equilateral: all equal) or by angles (acute: all angles <90°, right: one 90° angle, obtuse: one angle >90°). Understanding these properties helps verify solutions. For example, in a 3-4-5 triangle, the longest side (5) is opposite the largest angle (90°).
The Law of Cosines
The Law of Cosines relates the sides and angles: c² = a² + b² - 2ab·cos(C). It generalizes the Pythagorean theorem (when C=90°, cos(C)=0, reducing to c²=a²+b²). It's used in two scenarios: (1) SSS method: when all three sides are known, rearrange to find angles: cos(C) = (a²+b²-c²)/(2ab), then C = arccos(...). (2) SAS method: when two sides and the included angle are known, use the formula directly to find the third side. Example: If a=5, b=7, C=60°, then c² = 25 + 49 - 2(5)(7)cos(60°) = 74 - 70(0.5) = 39, so c ≈ 6.24. The Law of Cosines works for any triangle shape.
The Law of Sines
The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C). This means the ratio of each side to the sine of its opposite angle is constant for a given triangle. It's primarily used in ASA (angle-side-angle) and AAS (angle-angle-side) methods. When you know two angles and one side, first find the third angle (since angles sum to 180°), then use the Law of Sines to find the other sides. Example: If A=30°, B=60°, c=10, then C=90°. Using a/sin(A) = c/sin(C), we get a = 10·sin(30°)/sin(90°) = 10·(0.5)/1 = 5. Similarly, b = 10·sin(60°)/sin(90°) ≈ 8.66. The Law of Sines is simpler than Law of Cosines when angles are known.
Calculating Triangle Area
This calculator uses two main area formulas depending on available information. Heron's Formula (for SSS): When all three sides are known, first calculate the semi-perimeter s = (a+b+c)/2, then Area = √[s(s-a)(s-b)(s-c)]. Example: For a 3-4-5 triangle, s=6, Area = √[6·3·2·1] = 6. Trigonometric Formula (for SAS/ASA): When you know two sides and the included angle, Area = (1/2)·a·b·sin(C). Example: If a=6, b=8, C=90°, Area = (1/2)·6·8·sin(90°) = 24. This formula is computationally faster when an angle is known. Both methods give identical results for the same triangle, providing a way to verify calculations.
Frequently Asked Questions
When should I use SSS, SAS, or ASA methods?
Use SSS (side-side-side) when you know all three side lengths (e.g., a=3, b=4, c=5). Use SAS (side-angle-side) when you know two sides and the angle between them (e.g., a=5, C=90°, b=12). Use ASA (angle-side-angle) when you know two angles and the side between them (e.g., A=30°, c=10, B=60°). Each method requires specific information to uniquely determine the triangle.
What is the Law of Cosines and when is it used?
The Law of Cosines states: c² = a² + b² - 2ab·cos(C). It's used in SSS method to find angles when all sides are known, and in SAS method to find the third side when two sides and the included angle are known. It generalizes the Pythagorean theorem (when C=90°, cos(C)=0, giving c²=a²+b²). For example, with sides a=3, b=4, c=5, you can find angle C: cos(C) = (9+16-25)/(2·3·4) = 0, so C=90°.
What is the Law of Sines and how does it work?
The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C). It means the ratio of each side to the sine of its opposite angle is constant. It's primarily used in ASA and AAS methods. For example, if A=30°, B=60°, and c=10, first find C=90° (since angles sum to 180°), then use a = c·sin(A)/sin(C) = 10·sin(30°)/sin(90°) = 10·0.5/1 = 5.
How is triangle area calculated?
The calculator uses different formulas based on the method: For SSS, it uses Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s=(a+b+c)/2 is the semi-perimeter. For SAS and ASA, it uses Area = (1/2)·a·b·sin(C), which is more direct when an angle is known. For example, a triangle with sides 3, 4, 5 has s=6 and Area = √[6·3·2·1] = √36 = 6.
What does 'invalid triangle' mean?
A triangle is invalid if it violates the triangle inequality theorem: the sum of any two sides must be greater than the third side. For example, sides 1, 2, 5 don't form a valid triangle because 1+2=3 is not greater than 5. Also, angles must be positive and sum to exactly 180°. The calculator validates these conditions before solving.
How accurate is this triangle solver?
This calculator uses JavaScript's built-in trigonometric functions with double-precision floating-point arithmetic (15-17 significant digits). Results are displayed to 4 decimal places for sides and 2 decimal places for angles. For educational purposes and typical engineering calculations, this accuracy is excellent.
Is this tool free to use?
Yes! Completely free with no hidden costs, subscriptions, or limitations. Solve as many triangles as you need for homework, projects, or professional work.