Find The Degrees of A Triangle with Three Sides Calculator
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Reviewed by Calculator Editorial Team
This calculator helps you determine the angles of a triangle when you know the lengths of all three sides. It uses the Law of Cosines to find each angle based on the side lengths. The calculator provides precise angle measurements in degrees, helping you analyze triangle properties in geometry, engineering, and other applications.
How to Use This Calculator
To find the degrees of a triangle with three sides:
- Enter the length of side A in the first input field.
- Enter the length of side B in the second input field.
- Enter the length of side C in the third input field.
- Click the "Calculate" button to compute the angles.
- View the results showing the three angles of the triangle in degrees.
The calculator will display each angle with two decimal places for precision. You can reset the form at any time using the "Reset" button.
Formula Used
Law of Cosines
The Law of Cosines 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, the formula for angle A is:
cos(A) = (b² + c² - a²) / (2bc)
Similarly, the formulas for angles B and C are:
cos(B) = (a² + c² - b²) / (2ac)
cos(C) = (a² + b² - c²) / (2ab)
After calculating each cosine value, the angle in degrees is found using the arccosine function: angle = arccos(cosine value).
This formula works for any triangle, including right-angled, acute, and obtuse triangles. The calculator applies this formula to each angle based on the side lengths you provide.
Worked Example
Let's calculate the angles of a triangle with sides a = 5, b = 6, and c = 7.
- Calculate angle A:
cos(A) = (6² + 7² - 5²) / (2 * 6 * 7) = (36 + 49 - 25) / 84 = 60 / 84 ≈ 0.7143
A ≈ arccos(0.7143) ≈ 44.41°
- Calculate angle B:
cos(B) = (5² + 7² - 6²) / (2 * 5 * 7) = (25 + 49 - 36) / 70 = 38 / 70 ≈ 0.5429
B ≈ arccos(0.5429) ≈ 57.10°
- Calculate angle C:
cos(C) = (5² + 6² - 7²) / (2 * 5 * 6) = (25 + 36 - 49) / 60 = 12 / 60 = 0.2
C ≈ arccos(0.2) ≈ 78.46°
The angles of the triangle are approximately 44.41°, 57.10°, and 78.46°.
Note
The sum of the angles in any triangle is always 180°. In this example, 44.41° + 57.10° + 78.46° ≈ 180°, confirming the calculation is correct.
Frequently Asked Questions
What is the Law of Cosines?
The Law of Cosines is a theorem in Euclidean geometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It extends the Pythagorean theorem to non-right-angled triangles.
Can this calculator handle any type of triangle?
Yes, this calculator can handle any type of triangle, including right-angled, acute, and obtuse triangles, as long as the side lengths form a valid triangle.
What if the side lengths don't form a valid triangle?
The calculator will display an error message if the side lengths violate the triangle inequality theorem (the sum of any two sides must be greater than the third side).