How to Calculate the Length of a Curved Line

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A curved line, also called an "arc," represents a portion of a circle. It is difficult to measure a curve with a straight-edged ruler with any kind of accuracy, but geometry provides a relatively simple way to calculate the length of an arc. You'll need a tool called a protractor and some basic information. You must also know the diameter of the circle. Then, you can apply the following formula: length of an arc = diameter x 3.14 x the angle divided by 360.

TL;DR (Too Long; Didn't Read)

Remember that pi equals 3.14.

  1. Determine Diameter of Circle

  2. Determine diameter of the larger circle containing the arc. If you have the radius as a given, multiply that number by 2. For example, a radius of 5 inches equals a diameter of 10 inches

  3. Position Protractor to Measure Arc Angle

  4. Determine the angle of the arc by centering the protractor on the center point of the circle. The flat line at the bottom of the protractor called the "zero edge" must overlay the radius line and the zero degree mark on the protractor must overlay the bottom point of the arc.

  5. Determine Angle Degrees

  6. Note where the top point of the arc meets the protractor's degree scale. Wherever the arc ends defines the angle. For example, if the top point of the arc matches up to the 40 degree mark, your angle equals 40 degrees.

  7. Multiply Diameter by Pi and Arc Angle

  8. Multiply the diameter by 3.14 and then by the angle. In the examples used above with a diameter of 10 inches. and an angle of 40 degrees, you would use the following equation: 10 x 3.14 x 40, which equals 1256.

  9. Divide by Total Degrees

  10. Divide this product by 360 since there are 360 total degrees in a circle. In our example, this would be 1256 divided by 360 which equals 3.488.

  11. Round Decimal Result

  12. Round up the decimal if necessary to define the length of the arc. In our example, you could call the arc 3.49 inches if you round to hundredths or 3.5 inches if you round to tenths.

    Tips

    • Remember that the length of the arc is measured in the same units as the diameter. In this example, we use inches, but if the diameter were in centimeters, then the length of the arc would be 3.5 cm. If you are working on a practical problem, especially on a large scale, and have no way to determine diameter and angle, there is a simpler way. Lay out a string along the curve and cut it so that it lays perfectly on the curve. Then, measure the string.

References

About the Author

Sean Kotz has been writing professionally since 1988 and is a regular columnist for the Roanoke Times. He has also written for the Blue Ridge Business Journal, The Roanoker, 50 Plus, and Prehistoric Times, among others. He holds a Master of Arts in literature from Virginia Tech.

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