How to Calculate Angle of Triangle

Triangle Angle Calculator (SSS)

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

Calculated Angles:

Angle A:

Angle B:

Angle C:

(Sum of angles: ${sumOfAngles.toFixed(2)} degrees)

Understanding Triangle Angles: A Comprehensive Guide

Triangles are fundamental shapes in geometry, defined by three straight sides and three interior angles. A crucial property of any triangle is that the sum of its interior angles always equals 180 degrees. This principle is vital for solving many geometric problems and is the basis for calculating unknown angles.

What is an Angle of a Triangle?

An angle in a triangle is formed by the intersection of two sides. Each triangle has three such angles, typically denoted by capital letters (A, B, C) corresponding to the vertices, with the side opposite each angle denoted by its lowercase counterpart (a, b, c).

Methods to Calculate Triangle Angles

1. When Two Angles are Known

This is the simplest scenario. If you know the measure of two angles in a triangle, you can easily find the third using the fundamental rule:

Angle A + Angle B + Angle C = 180°

So, if you know Angle A and Angle B, then Angle C = 180° - (Angle A + Angle B).

Example: If Angle A = 60° and Angle B = 70°, then Angle C = 180° – (60° + 70°) = 180° – 130° = 50°.

2. When Three Sides are Known (SSS – Side-Side-Side) – Using the Law of Cosines

When you know the lengths of all three sides of a triangle (a, b, and c), you can calculate each of its interior angles using the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles.

The formulas derived from the Law of Cosines are:

• To find Angle A (opposite side 'a'):
  cos(A) = (b² + c² - a²) / (2bc)
• To find Angle B (opposite side 'b'):
  cos(B) = (a² + c² - b²) / (2ac)
• To find Angle C (opposite side 'c'):
  cos(C) = (a² + b² - c²) / (2ab)

After calculating the cosine value, you take the inverse cosine (arccos or cos⁻¹) to find the angle in radians, then convert it to degrees by multiplying by 180/π.

Example: Consider a triangle with sides a = 5, b = 7, and c = 10.

• For Angle A:
  cos(A) = (7² + 10² - 5²) / (2 * 7 * 10)
cos(A) = (49 + 100 - 25) / 140
cos(A) = 124 / 140 ≈ 0.8857
A = arccos(0.8857) ≈ 27.66°
• For Angle B:
  cos(B) = (5² + 10² - 7²) / (2 * 5 * 10)
cos(B) = (25 + 100 - 49) / 100
cos(B) = 76 / 100 = 0.76
B = arccos(0.76) ≈ 40.54°
• For Angle C:
  cos(C) = (5² + 7² - 10²) / (2 * 5 * 7)
cos(C) = (25 + 49 - 100) / 70
cos(C) = -26 / 70 ≈ -0.3714
C = arccos(-0.3714) ≈ 111.80°

Sum of angles: 27.66° + 40.54° + 111.80° = 180.00°. This confirms our calculations.

3. When Two Sides and the Included Angle are Known (SAS – Side-Angle-Side)

If you know two sides and the angle between them, you can first use the Law of Cosines to find the third side. Once all three sides are known, you can then use the Law of Cosines (as in method 2) or the Law of Sines to find the remaining angles.

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Using the Triangle Angle Calculator

Our calculator above simplifies the process for the SSS scenario. Simply input the lengths of the three sides (a, b, and c) into the respective fields. The calculator will then apply the Law of Cosines to determine each of the interior angles (A, B, and C) and display them in degrees. It also performs a check to ensure the entered side lengths can actually form a valid triangle, based on the Triangle Inequality Theorem (the sum of the lengths of any two sides of a triangle must be greater than the length of the third side).

Important Considerations

• Units: While the calculator doesn't require specific units for side lengths (e.g., cm, inches), ensure consistency. All three side lengths should be in the same unit. The angles will always be in degrees.
• Triangle Inequality: Always remember that for a triangle to exist, the sum of any two sides must be greater than the third side. If this condition is not met, the calculator will inform you that a valid triangle cannot be formed.
• Right Triangles: For right-angled triangles, you can also use trigonometric ratios (SOH CAH TOA – Sine, Cosine, Tangent) if you know two sides or one side and one angle. However, the Law of Cosines works for all types of triangles, including right-angled ones.

Whether you're a student, an engineer, or just curious about geometry, understanding how to calculate triangle angles is a valuable skill. Our calculator provides a quick and accurate way to perform these calculations for any triangle where all three side lengths are known.