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How to Find the Radius of a Curvature

by William Nagel, Demand Media

    The radius of a curvature is the radius of a circle drawn through parts of a curve. This radius can be used for a variety of mechanical, physical and optical calculations. Finding the radius requires the use of calculus. The formula for finding the radius of a curvature is: {[1+(dy/dx)^2]^3/2} / |d^2y/dx^2| To calculate the radius of a curvature, take the equation of your curve and use the radius of a curvature formula to solve for a variable “x” at a point along the curve.

    Items you will need

    • Pencil
    • Paper
    • Calculator
    • Equation of a curve
    Step 1

    Calculate the derivative, dy/dx, of your curve. Using this result, calculate the second derivative, d^2y/dx,

    Step 2

    Square the first derivative, dy/dx, and plug the result into the formula for finding the radius of a curvature. Put the result into the formula at (dy/dx)^2.

    Step 3

    Plug the second derivative of your curve equation into the formula for finding the radius of a curvature. Put the second derivative into the formula at d^2y/dx^2.

    Step 4

    Solve the equation for a point “x” along your curve by replacing the variable "x" with a numerical value. Use a calculator to speed your calculations.

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    Tip

    • Some advanced graphing calculators have a built-in function that will calculate the radius of a curvature automatically. If you have a graphing calculator with this function, use it to check your work.

    Warning

    • Always check your work to make sure it is accurate.

    About the Author

    William Nagel is a graduate of the Georgia Institute of Technology, where he studied science, technology and culture. He has been writing since 2007 and specializes in computer hardware, operating systems and software documentation. His work has been published in the "North Avenue Review."

    Photo Credits

    • NA/AbleStock.com/Getty Images

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