Triangle Degree
Find all three interior angles of any triangle from its three side lengths using the law of cosines. Free online geometry calculator with instant degree results, triangle type classification, and angle composition chart.
About This Calculator
This free online triangle degree calculator finds all three interior angles of any triangle when you know the lengths of all three sides. Using the law of cosines, it computes angles A (alpha), B (beta), and C (gamma) in degrees with high precision. The calculator also validates whether the three side lengths satisfy the triangle inequality theorem, classifies the triangle as acute, right, or obtuse based on the largest angle, and displays an interactive angle composition chart.
The law of cosines is a fundamental rule in trigonometry that extends the Pythagorean theorem to any triangle. For a triangle with sides a, b, c and opposite angles A, B, C, the formulas are: A = arccos((b^2 + c^2 - a^2) / (2bc)), B = arccos((a^2 + c^2 - b^2) / (2ac)), and C = arccos((a^2 + b^2 - c^2) / (2ab)). This calculator automates these computations, giving you instant results without manual trigonometric calculations.
Before computing the angles, the calculator applies the triangle inequality theorem to verify that the three side lengths can form a valid triangle. The sum of any two sides must exceed the third side: a + b > c, a + c > b, and b + c > a. If the sides fail this check, the calculator clearly indicates an invalid triangle. For valid triangles, it automatically classifies the triangle type: acute (all angles less than 90 deg), right (one angle exactly 90 deg), or obtuse (one angle greater than 90 deg).
This triangle degree calculator is ideal for geometry students, trigonometry learners, and professionals in engineering, architecture, construction, and design who need quick and accurate angle measurements. Whether you're verifying homework, designing a roof truss, analyzing a structural frame, or simply exploring triangle geometry, this tool delivers reliable results instantly. The angle composition chart provides a visual representation of how the three angles compare, making it easier to understand the triangle's proportions at a glance.
The concepts behind this calculator -- the law of cosines and triangle inequality -- are universally taught in mathematics curricula worldwide. Students in India (CBSE, ICSE), the United States (Common Core), and the United Kingdom (GCSE, A-Level) all learn these principles as part of their geometry and trigonometry studies. The calculator works with any unit of measurement (centimeters, meters, inches, feet) since angles depend only on the relative side proportions, not the absolute units.
Frequently Asked Questions
How do I find the angles of a triangle if I know all three sides?
Use the law of cosines: angle A = arccos((b^2 + c^2 - a^2) / (2bc)), angle B = arccos((a^2 + c^2 - b^2) / (2ac)), and angle C = arccos((a^2 + b^2 - c^2) / (2ab)). Simply enter your three side lengths into the triangle degree calculator and it will compute all three angles instantly.
What is the law of cosines for finding triangle angles?
The law of cosines relates the side lengths of a triangle to the cosine of one of its angles. For a triangle with sides a, b, c and opposite angles A, B, C, the formula is: a^2 = b^2 + c^2 - 2bc·cos(A), which rearranges to A = arccos((b^2 + c^2 - a^2) / (2bc)). This formula works for any triangle -- acute, right, or obtuse.
How can I determine if three side lengths form a valid triangle?
Three side lengths form a valid triangle if they satisfy the triangle inequality theorem: the sum of any two sides must be greater than the third side (a + b > c, a + c > b, and b + c > a). If all three conditions are met, a unique triangle exists. The triangle degree calculator automatically checks this and will indicate if the sides form an invalid triangle.
What types of triangles can this calculator identify?
The calculator classifies triangles into three types based on the largest angle: acute triangles have all angles less than 90 deg, right triangles have one angle exactly 90 deg, and obtuse triangles have one angle greater than 90 deg. It also detects invalid triangles that do not satisfy the triangle inequality theorem.
Can I use this calculator for right triangles specifically?
Yes, the triangle degree calculator works perfectly for right triangles. Enter the three side lengths and the calculator will determine all angles. If one angle is exactly 90 deg, it will be classified as a right triangle. For a dedicated right triangle solver, check our right triangle calculator for side and area calculations.
What if I only know two angles of a triangle?
If you know two angles of a triangle, you can find the third angle using the triangle sum theorem: subtract the sum of the two known angles from 180 deg. Our triangle angle calculator handles this specific use case. The triangle degree calculator here is designed for when you know all three side lengths.
Is this calculator useful for students and professionals?
Absolutely. Students learning geometry, trigonometry, or preparing for exams (CBSE, ICSE, Common Core, GCSE) can verify their angle calculations. Engineers, architects, carpenters, and construction professionals use it to find angles in structures, trusses, and designs without manual computation.
How accurate are the angle results?
The calculator uses the standard law of cosines formula with JavaScript's Math.acos function, providing results accurate to at least 6 decimal places. Results are displayed rounded to 2 decimal places for readability. The sum of all three angles will always equal 180 deg (within floating-point rounding tolerance) for any valid triangle.