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:

    What is Law of Cosines?

    The Law of Cosines, also known as the Cosine Rule, is a fundamental relationship in trigonometry that connects the lengths of the sides of a triangle to the cosine of one of its angles. It’s essentially an extension of the Pythagorean Theorem, which only works for right triangles, to triangles of any shape.

    Key Aspects of Law of Cosines

    • Generalizes the Pythagorean Theorem: While the Pythagorean Theorem only works for right triangles, the Law of Cosines applies to triangles of any shape. It’s like a more versatile toolbox for triangles!
    • Formula: The basic formula relates the square of a side (c) to the squares of the other two sides (a and b) and the cosine of the angle (C) between them: c^2 = a^2 + b^2 - 2ab cos(C). There are similar formulas for finding the other sides.
    • Finding Missing Sides: If you know two sides and the angle between them, you can use the Law of Cosines to find the missing third side. This is handy in situations like surveying land or navigating with triangulation.
    • Finding Missing Angles: Knowing all three side lengths, you can use the Law of Cosines to calculate any of the angles. This can be useful in structural analysis or designing objects with specific angles.

    All Formulae Related to Law of Cosines Calculator

    Here are the formulae related to the Law of Cosines calculator:

    Core Formula:

    • c^2 = a^2 + b^2 – 2ab cos(C)

    Variations for finding other sides:

    • a^2 = b^2 + c^2 – 2bc cos(A)
    • b^2 = a^2 + c^2 – 2ac cos(B)

    Formulae for finding angles:

    • cos(A) = (b^2 + c^2 – a^2) / 2bc
    • cos(B) = (a^2 + c^2 – b^2) / 2ac
    • cos(C) = (a^2 + b^2 – c^2) / 2ab

    Applications of Law of Cosines Calculator in Various Fields

    Navigation and Surveying:

    • Trilateration: Imagine yourself lost in a forest. You stumble upon three radio towers with known locations. By measuring the angles between these towers from your position, you can use the Law of Cosines calculator to triangulate your exact coordinates. Similar principles are used in land surveying and ship navigation.
    • Satellite positioning: Global Positioning Systems (GPS) rely heavily on trigonometry. When your phone receives signals from multiple satellites, the Law of Cosines calculator helps determine your precise location based on the angles and distances involved.

    Engineering and Architecture:

    • Bridge construction: Building sturdy bridges requires precise calculations to handle immense weight and stress. The Law of Cosines calculator helps engineers determine the forces acting on different bridge components due to angles and weight distribution, ensuring the bridge’s structural integrity.
    • Building design: Architects incorporate triangular elements in buildings for both aesthetic and structural purposes. The Law of Cosines calculator becomes crucial in determining the dimensions and angles of these elements to maintain stability and achieve the desired design.

    Astronomy and Space Exploration:

    • Stellar distances: Astronomers measure the vast distances between stars using a technique called parallax. By observing a star from different positions on Earth and measuring the tiny shifts in its apparent position (parallax angle), the Law of Cosines calculator helps calculate the star’s actual distance from Earth.
    • Rocket orbits: Launching rockets to specific destinations requires precise trajectories. The Law of Cosines calculator helps determine the angles and thrust needed to achieve the desired orbital path, ensuring a successful mission.

    Other Applications:

    • Robotics: Robots rely on precise calculations for joint movements and object manipulation. The Law of Cosines calculator helps programmers define these movements based on angles and distances, enabling robots to perform complex tasks efficiently.
    • Sports analysis: Analyzing the trajectories of balls in sports like football or baseball involves trigonometry. The Law of Cosines calculator can predict the path of a thrown ball or calculate the bounce angle off a surface, helping athletes and coaches optimize their performance.

    Benefits of Using the Law of Cosines Calculator

    Efficiency and Accuracy:

    • Saves time: Manual calculations with the Law of Cosines formula can be tedious and prone to errors. The calculator performs the necessary steps quickly and accurately, freeing you up for other tasks.
    • Reduces errors: Complex calculations involving trigonometric functions have a higher risk of human error. The calculator eliminates this risk and ensures reliable results.

    Accessibility and Ease of Use:

    • Simple interface: Most calculators offer user-friendly interfaces, requiring only the input of known side lengths and angles. Even those unfamiliar with the formula can utilize it effectively.
    • Visualization: Some calculators provide visual representations of the triangle and its measurements, which can be helpful for understanding the problem and interpreting the results.

    Versatility and Flexibility:

    • Solve various problems: The calculator can handle different scenarios, whether you’re finding missing sides, angles, or verifying existing measurements.
    • Experimentation: You can easily explore different configurations and see how changes in one variable affect the others, which can be valuable for problem-solving and gaining deeper insights.

    Learning and Understanding:

    • Verification tool: The calculator can be used to check your own manual calculations and reinforce your understanding of the Law of Cosines.
    • Exploration tool: Visualizing the relationships between sides and angles in different scenarios can deepen your understanding of the concept and its practical applications.



    Last Updated : 27 February, 2024

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    14 thoughts on “Law of Cosines Calculator”

    1. This article provides a comprehensive and detailed explanation of the Law of Cosines, it’s a great resource!

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