Find Angle of Triangle Calculator

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Triangle Angle Calculator

Enter the lengths of the three sides of a triangle to calculate its interior angles.

Understanding Triangle Angles and the Law of Cosines

A triangle is a fundamental shape in geometry, defined by three straight sides and three interior angles. The sum of these interior angles always equals 180 degrees. This calculator helps you determine the measure of each angle when you know the lengths of all three sides.

The Law of Cosines

To find the angles of a triangle given its three sides (often referred to as the SSS case – Side-Side-Side), we use 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 the angles, we rearrange these formulas:

  • cos(A) = (b² + c² - a²) / (2bc)
  • cos(B) = (a² + c² - b²) / (2ac)
  • cos(C) = (a² + b² - c²) / (2ab)

Once you have the cosine of an angle, you can find the angle itself by taking the inverse cosine (arccos or cos⁻¹) of that value. The result will typically be in radians, which then needs to be converted to degrees (1 radian = 180/π degrees).

Triangle Inequality Theorem

It's important to remember that not any three arbitrary lengths can form a triangle. For three segments to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the Triangle Inequality Theorem:

  • a + b > c
  • a + c > b
  • b + c > a

If these conditions are not met, the calculator will indicate that the sides cannot form a valid triangle.

Example Calculation

Let's say you have a triangle with sides:

  • 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 our calculations. This specific example is a right-angled triangle, as indicated by the 90-degree angle.

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