# 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.