Triangle Angle Calculator
Use this calculator to determine the interior angles of a triangle when you know the lengths of all three sides. This tool applies the Law of Cosines to accurately find each angle in degrees.
Calculate Angles
Understanding Triangle Angles
A triangle is a fundamental polygon with three sides and three interior angles. The sum of the interior angles of any triangle always equals 180 degrees. Knowing the angles of a triangle is crucial in various fields, from geometry and trigonometry to engineering, architecture, and even computer graphics.
The Law of Cosines
When you know the lengths of all three sides of a triangle (often referred to as the SSS case – Side-Side-Side), you can determine each of its interior angles using the Law of Cosines. This law is a generalization of the Pythagorean theorem and relates the lengths of the sides of a triangle to the cosine of one of its angles.
For a triangle with sides a, b, and c, and angles A, B, and C opposite those respective sides, the Law of Cosines states:
c² = a² + b² - 2ab * cos(C)b² = a² + c² - 2ac * cos(B)a² = b² + c² - 2bc * cos(A)
To find an angle, we rearrange these formulas:
- Angle A:
A = arccos((b² + c² - a²) / (2bc)) - Angle B:
B = arccos((a² + c² - b²) / (2ac)) - Angle C:
C = arccos((a² + b² - c²) / (2ab))
The arccos function (also known as inverse cosine) returns the angle whose cosine is the given value. The result is typically in radians, which then needs to be converted to degrees for practical use (1 radian = 180/π degrees).
How to Use the Calculator
- Enter Side Lengths: Input the numerical values for the lengths of Side A, Side B, and Side C into the respective fields. Ensure these are positive numbers.
- Click Calculate: Press the "Calculate Angles" button.
- View Results: The calculator will display the calculated values for Angle A, Angle B, and Angle C in degrees. It will also check if the entered side lengths can form a valid triangle.
Example Calculation
Let's consider a triangle with the following side lengths:
- Side A = 3 units
- Side B = 4 units
- Side C = 5 units
Using the Law of Cosines:
For Angle A:
cos(A) = (4² + 5² - 3²) / (2 * 4 * 5)
cos(A) = (16 + 25 - 9) / 40
cos(A) = 32 / 40 = 0.8
A = arccos(0.8) ≈ 36.87 degrees
For Angle B:
cos(B) = (3² + 5² - 4²) / (2 * 3 * 5)
cos(B) = (9 + 25 - 16) / 30
cos(B) = 18 / 30 = 0.6
B = arccos(0.6) ≈ 53.13 degrees
For Angle C:
cos(C) = (3² + 4² - 5²) / (2 * 3 * 4)
cos(C) = (9 + 16 - 25) / 24
cos(C) = 0 / 24 = 0
C = arccos(0) = 90 degrees
The sum of the angles is 36.87 + 53.13 + 90 = 180 degrees, confirming a valid triangle (specifically, a right-angled triangle).