Law of Cosines Calculator

Law of Cosines Calculator

  • Enter the values for Side A, Side B, and Angle C.
  • Select the appropriate units for each value.
  • Click the "Calculate" button to perform the calculations.
  • View the results including Side C, Angle A, Angle B, Area, and Semiperimeter.
  • Copy the results to the clipboard using the "Copy Results" button.
  • Your calculation history is displayed below.

Side C:

Angle A:

Angle B:



Calculation Details:
Calculation History:


    The Law of Cosines Calculator is a valuable mathematical tool used to solve triangles when the three sides and one angle, or two sides and two angles are known. This calculator employs the Law of Cosines, a fundamental trigonometric concept that extends the Pythagorean theorem to non-right triangles.

    The Law of Cosines Formula

    The Law of Cosines is a mathematical formula used to find the measures of the angles and sides of a triangle when certain information is known. The formula is as follows:

    c² = a² + b² – 2ab * cos(C)


    • c represents the length of the side opposite the angle C.
    • a and b denote the lengths of the other two sides.
    • C is the measure of the angle opposite side c.
    • cos(C) is the cosine of angle C.

    This formula allows us to solve for any of the three sides or any of the three angles within a triangle, given that we know the values of at least three of these parameters.

    Example Calculations

    Example 1: Finding an Angle

    Suppose we have a triangle with side lengths a = 4 units, b = 5 units, and c = 6 units. We want to find the measure of angle C. Using the Law of Cosines:

    6² = 4² + 5² – 2 * 4 * 5 * cos(C)


    36 = 16 + 25 – 40 * cos(C)

    Combine like terms:

    36 = 41 – 40 * cos(C)

    Rearrange the equation:

    40 * cos(C) = 41 – 36

    40 * cos(C) = 5

    Now, isolate cos(C):

    cos(C) = 5 / 40

    cos(C) = 1/8

    Finally, find the angle C by taking the inverse cosine:

    C = cos⁻¹(1/8)

    C ≈ 82.82 degrees

    Example 2: Finding a Side Length

    Consider a triangle with angles A = 30 degrees, B = 45 degrees, and side length c = 8 units. We want to find the length of side a. Using the Law of Cosines:

    a² = b² + c² – 2bc * cos(A)

    Substituting known values:

    a² = b² + 8² – 2 * 8 * 8 * cos(30)

    a² = b² + 64 – 128 * (sqrt(3)/2)

    a² = b² + 64 – 64 * sqrt(3)

    Now, if we assume b = 6 units:

    a² = 6² + 64 – 64 * sqrt(3)

    a² = 100 – 64 * sqrt(3)

    a ≈ 4.14 units

    Real-World Use Cases

    The Law of Cosines and its calculator are essential in various real-world scenarios, such as:

    1. Navigation: In land and maritime navigation, determining distances and angles between points on Earth’s surface is crucial. The Law of Cosines helps calculate great circle distances accurately.
    2. Engineering: Engineers use the Law of Cosines to analyze and design structures, such as truss bridges or antennas, where non-right triangles are prevalent.
    3. Physics: In physics, the calculator is applied to analyze vector forces acting on an object in two or three dimensions.
    4. Astronomy: Astronomers use the Law of Cosines to calculate angular separations between celestial objects and determine their positions.
    5. Geography: Geographers use it to measure distances on maps and determine the shape of land masses accurately.


    The Law of Cosines Calculator is a versatile tool that plays a vital role in solving triangles and a wide range of applications. By understanding the Law of Cosines formula and its applications, one can navigate, engineer, and analyze various real-world situations with precision.


    1. Stewart, James. (2019). “Calculus: Early Transcendentals.” Cengage Learning.
    2. Stroud, K. A., & Booth, D. J. (2013). “Engineering Mathematics.” Palgrave Macmillan.
    dot 1
    One request?

    I’ve put so much effort writing this blog post to provide value to you. It’ll be very helpful for me, if you consider sharing it on social media or with your friends/family. SHARING IS ♥️

    Leave a Comment

    Your email address will not be published. Required fields are marked *