Triangle Angle Calculator
Free Triangle angle Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Formula
Angle Sum: A + B + C = 180 | Law of Cosines: cos(A) = (b^2 + c^2 - a^2) / (2bc)
The angle sum property states all triangle angles total 180 degrees. The Law of Cosines finds any angle from three known sides by relating the cosine of an angle to the lengths of all three sides.
Worked Examples
Example 1: Finding the Third Angle from Two Known Angles
Problem: A triangle has angles A = 55 degrees and B = 72 degrees. Find angle C and classify the triangle.
Solution: Using the angle sum property:\nAngle C = 180 - A - B\nAngle C = 180 - 55 - 72 = 53 degrees\n\nAll angles are less than 90 degrees, so this is an Acute triangle.\nAll angles are different, so this is a Scalene triangle.\n\nIn radians: A = 0.9599, B = 1.2566, C = 0.9250\nVerification: 0.9599 + 1.2566 + 0.9250 = 3.1416 = pi
Result: Angle C = 53 degrees | Acute Scalene Triangle
Example 2: Finding All Angles from Three Sides
Problem: A triangle has sides a = 6, b = 8, c = 10. Find all three angles.
Solution: Using the Law of Cosines:\ncos(A) = (8^2 + 10^2 - 6^2) / (2 x 8 x 10) = (64 + 100 - 36) / 160 = 128/160 = 0.8\nA = arccos(0.8) = 36.87 degrees\n\ncos(B) = (6^2 + 10^2 - 8^2) / (2 x 6 x 10) = (36 + 100 - 64) / 120 = 72/120 = 0.6\nB = arccos(0.6) = 53.13 degrees\n\nC = 180 - 36.87 - 53.13 = 90.00 degrees\n\nThis is a Right triangle (3-4-5 Pythagorean triple scaled by 2)
Result: A = 36.87 deg | B = 53.13 deg | C = 90.00 deg | Right Scalene Triangle
Frequently Asked Questions
What is the triangle angle sum property?
The triangle angle sum property states that the three interior angles of any triangle always add up to exactly 180 degrees (or pi radians). This is one of the most fundamental theorems in Euclidean geometry and applies to all triangles regardless of their shape, size, or type. The proof follows from drawing a line through one vertex parallel to the opposite side and using alternate interior angles. This property means that if you know any two angles of a triangle, you can always find the third by subtracting their sum from 180. It also means no triangle can have more than one right angle (90 degrees) or more than one obtuse angle (greater than 90 degrees). This property does not hold in non-Euclidean geometries like spherical geometry.
How do you find the third angle when two angles are known?
Finding the third angle of a triangle is straightforward when two angles are known. Simply subtract the sum of the two known angles from 180 degrees. For example, if angle A = 45 degrees and angle B = 75 degrees, then angle C = 180 - 45 - 75 = 60 degrees. This works because the angle sum property guarantees all three angles total exactly 180 degrees. In radians, the process is identical but you subtract from pi instead of 180. If angle A = pi/4 and angle B = pi/3, then angle C = pi - pi/4 - pi/3 = pi - 3pi/12 - 4pi/12 = 5pi/12 radians. Always verify that each angle is positive and that none exceeds 180 degrees, as this would indicate an invalid triangle.
What is the exterior angle theorem?
The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. An exterior angle is formed by extending one side of the triangle beyond a vertex. For example, if a triangle has angles 40, 60, and 80 degrees, the exterior angle at the 80-degree vertex equals 40 + 60 = 100 degrees. This makes sense because the exterior angle and its adjacent interior angle are supplementary (sum to 180), so the exterior angle = 180 - adjacent interior angle = sum of the other two angles. This theorem is extremely useful in geometry proofs and problem-solving because it establishes a relationship between interior and exterior angles without needing to know all three interior angles directly.
Can I use Triangle Angle Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Can I share or bookmark my calculation?
You can bookmark the calculator page in your browser. Many calculators also display a shareable result summary you can copy. The page URL stays the same so returning to it will bring you back to the same tool.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.