Triangle solver
Fast, privacy-first online Triangle solver tool. 100% in-browser client-side execution with zero signups and zero data tracking.
1. Known Sides & Angles
Geometric Visualizer
2. Solved Geometry & Step-by-Step
Step-by-Step Derivation
- Step 1 (Law of Cosines): Find angle A using cos(A) = (b² + c² - a²) / (2bc) = (36 + 49 - 25) / 84 = 0.7143 → A = 44.42°
- Step 2 (Law of Sines): Find angle B using sin(B) = (b × sin(A)) / a = (6 × 0.7000) / 5 = 0.8399 → B = 57.12°
- Step 3 (Angle Sum): C = 180° - (A + B) = 180° - (44.42° + 57.12°) = 78.46°
- Step 4 (Heron's Formula): s = (5 + 6 + 7) / 2 = 9. Area = √[9(4)(3)(2)] = √216 ≈ 14.70
About this tool
How to use it
- Select your known triangle properties from the configuration dropdown: SSS (3 sides), SAS (side-angle-side), ASA, AAS, or SSA.
- Input the numerical values for the known sides or angles in degrees.
- Inspect the real-time scaled SVG geometry diagram to visually verify vertices A, B, C and side lengths a, b, c.
- Review the calculated angle values (α, β, γ) in both degrees and radians.
- Check geometric metrics: total area (via Heron's formula), perimeter, semi-perimeter, and inscribed circle radius (r).
- Read the step-by-step mathematical derivation showing the exact Law of Sines and Law of Cosines operations.
Why it matters
Solving oblique non-right triangles requires multiple sequential steps using the Law of Cosines and Law of Sines, with careful handling of inverse trigonometric rounding and the ambiguous SSA case. An automated visual solver eliminates calculation errors and helps students understand the geometric proofs.
When to use it
Use when checking geometry and trigonometry homework, calculating land surveying boundary lines, designing woodworking roof trusses, or solving 3D computer graphics vector angles.
Good to know
In Euclidean geometry, the sum of all three interior angles in any planar triangle is always exactly 180 degrees (π radians). Furthermore, the Triangle Inequality theorem dictates that the sum of any two side lengths must be strictly greater than the third.
Common questions
What is the Law of Cosines?
The Law of Cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² - 2ab·cos(C). It is used to solve SSS and SAS triangle cases.
What is the Law of Sines?
The Law of Sines states that the ratio of each side length to the sine of its opposite angle is constant: a / sin(A) = b / sin(B) = c / sin(C). It is used to solve ASA, AAS, and SSA cases.
What is the ambiguous case (SSA)?
When given two sides and a non-included angle (SSA), depending on the side lengths, there may be zero triangles, exactly one unique triangle, or two distinct valid triangles. Our solver identifies whether a valid solution exists.
How is the area of an oblique triangle calculated?
When all three sides are known, the solver uses Heron's formula: Area = √[s(s - a)(s - b)(s - c)], where s is the semi-perimeter (a + b + c) / 2. When two sides and an included angle are known, Area = ½ab·sin(C).
Does this tool work completely offline?
Yes. All trigonometric calculations and dynamic SVG rendering execute 100% locally in your web browser memory with zero network dependencies.